Upload
phungnhu
View
227
Download
0
Embed Size (px)
Citation preview
A IDENTITIES: TRIGONOMETRIC,
SECH AND TANH
"For me, the criterion of science is truth, but its motivation resides in a sense of beauty - and in that it is like art."
~ Chia-Shun Yih, (1918-1997)
A.I DIFFERENTIATION IDENTITIES
Sech & Tanh, Cosh & Sinh:
d dx sech(x) = -sech(x) tanh(x),
d 2 dx tanh(x) = sech (x) (A.l)
d . dx cosh(x) = smh(x),
d . dx smh(x) = cosh(x) (A.2)
Powers of Sech:
- jsechj(x) tanh(x)
j4 sechj (x) - 2j(j + 1)(j2 + 2j + 2) sechJ+2(x)
+j(j + 1)(j + 2)(j + 3)sechj+4(x) (A.3)
482
Trigonometric and Sech Identities 483
Tanh Times Powers of Sech:
d . dx {tanh(x)sechJ(x)}
d2 . dx2 {tanh(x) sechJ (x)} tanh(x) {j2 sech j (x) - (i + 3j + 2)sech1+2 (x) }
d3 . dx3 {tanh(x) sechJ (x)} - j3 sechj (x) + (2j3 + 6i + Sj + 4)sech1+2(x)
_(j3 + 6i + 11j + 6)sechj+4(x) (A.4)
Differentiation Chain:
f{ -1) - ~x dylog(cosh(y)) (A.5)
1 w2 1 2"X2 -log(2) y + 24 + 2"dilog (1 + exp( -2x))
f - log(cosh(x)) = x log(2) + log (1 + exp( -2x)) (A.6) df
tanh(x) (A.7) dx
d2f sech2(x) (A.S)
dx2 d3 f
-2 tanh(x) sech2(x) (A. g) dx3 d4f
4sech2(x) - 6sech4(x) (A.I0) dx4 d5 f
tanh(x) {-Ssech2(x) + 24sech4(x)} (A.11) dx 5
d6 f 16sech2(x) -120sech4(x) + 120sech6 (x) (A.12)
dx6
d7 f tanh(x) {-32sech2(x) + 4S0sech4(x) - 720sech6 (x)} (A.13)
dx7
484 ApPENDIX A
A.2 HYPERBOLIC IDENTITIES
Connections:
sech2(x) 1 - tanh2(x)
cosh2(x) - sinh2(x) 1
cosh2(X) + sinh2(x) = cosh(2x)
2 sinh(A) sinh(B)
2 cosh(A) cosh(B)
2 sinh(A) cosh(B)
cosh(A + B)
sinh(A + B) =
tanh(A + B) =
coth(A + B)
Products:
cosh(A + B) - cosh(A - B)
cosh(A + B) + cosh(A - B)
sinh(A + B) + sinh(A - B)
Addition:
cosh(A) cosh(B) + sinh(A) sinh(B)
sinh(A) cosh(B) + cosh(A) sinh(B)
tanh(A) + tanh(B) 1 + tanh(A) tanh(B) cot(A) cot (B) - 1 cot(A) + cot(B)
Logarithmic Form of Inverse Hyperbolic Functions:
arccosh(x) = log ( x + #'=1) arcsinh(x) = log ( x + vi x2 + 1) [, x2 2:: 1]
arcsech(x) = en log ;; + ;; - 1 [0 < x ::; 1 J
arctanh(x) 1 (I-X) [0 ::; x2 < IJ = - log --2 l+x
(A.14)
(A.15)
(A.16)
(A.17)
(A.18)
(A.19)
(A.20)
(A.21)
(A.22)
(A.23)
(A.24)
(A.25)
(A.26)
(A.27)
Trigonometric and Sech Identities
A.3 TRIGONOMETRIC IDENTITIES
Products: 2 sin(A) sin(B) cos(A - B) - cos(A + B)
2cos(A) cos(B) cos(A - B) + cos(A + B)
2 sin(A) cos(B) sin(A - B) + sin(A + B)
Addition:
cos(A + B) cos(A) cos(B) - sin(A) sin(B)
sin(A + B) sin(A) cos(B) + cos(A) sin(B)
tan(A + B) tan(A) + tan(B)
1 - tan(A) tan(B)
cot(A + B) cot(A) cot(B) - 1
cot(A) + cot(B)
Powers:
cos2 (x) 1 1 2 + 2 cos(2x)
cos3 (x) 3 1 "4 cos(x) + "4 cos(3x)
cos4 (x) 311 "8 + 2 cos(2x) + "8 cos(4x)
Trigonometric-to-Powers: sin(2x) 2 sin (x) cos(x)
sin(3x) 3sin(x) - 4sin3 (x)
sin(4x) -4 sin(x) cos(x) + 8 sin(x) cos3 (x)
cos(2x) = -1 + 2cos2 (x) = 1 - 2sin2 (x)
cos(3x) -3cos(x) + 4 cos3 (x)
cos(4x) 1 - 8cos2 (x) + 8cos4 (x)
sin (x + ~) = cos(x) , cos (x + ~) = - sin(x)
tan (x + ~) = - cot(x)
485
(A.28)
(A.29)
(A.30)
(A.31)
(A.32)
(A.33)
(A.34)
(A.35)
(A.36)
(A.37)
(A.38)
(A.39)
(A.40)
(A.41)
(A.42)
(A.43)
(A.44)
(A.45)
B SECH/TANH PERTURBATION
SERIES
"[I said to Sir Arthur Cayley, who never used quarternions:] 'You know, quarternions are just like a pocket-map.' He replied 'That may be, but you've got to take it out of your pocket, and unfold it, before it's of any use.' And he dismissed the subject with a smile."
- P. G. Tait, in the Life of Lord Kelvin by Silvanus Thompson, pg. 1037.
B.l POLYNOMIALIZATION
For many solitary wave problems, each order j of the multiple scales perturbation series is a polynomial in sech( EX) or such a polynomial multiplied by a single factor of tanh(EX). It is fairly straightforward to compute such series to high order in a numerical language like FORTRAN by using the identities of Appendix A plus a little hand analysis to derive recurrences.
Algebraic manipulation languages like Maple and Mathematica offer the hope of eliminating some of the paper-and-pencil analysis. However, these languages are much less adept at manipulating transcendentals than algebraic functions. The Maple simplification command, for example, will convert sech and tanh into exponentials, which is not what one wants at all.
One work-around is to note that all powers of sech and tanh can be converted into polynomials or algebraic functions by the change of coordinate
z == tanh(x) (B.l)
(In applications, x = EX, but we omit the dependence on E for notational simplicity.) The following identities are helpful in this conversion:
sech(x)
sech2(x)
486
(B.2)
(B.3)
Sech/Tanh Series 487
d dx
(1 - Z2) ! (B.4)
d2
dx2 2{ 2d2 d} (1 - z) (1 - z ) dz2 - 2z dz
d3
dx3 = 2 { 2)2 d3 2 d2 2 d} (1 - z) (1 - z dz3 - 6z(1 - z ) dz2 + (4 - 6(1 - z )) dz .
Maple and Mathematica can manipulate high degree polynomials, perhaps multiplied by a square root, very easily.
If the differential equation is translationally invariant in x, then the transformed equation will have polynomial coefficients in z. For FKdV and TNLS solitary waves, the terms in the perturbation series are polynomials in z (or polynomials multiplied by a square root of (1 - Z2)). However, in the RMKdV equation, the second order and higher orders are not polynomials.
Another illuminating counterexample is the Korteweg-deVries-Burgers equation. Its travelling shocks for unit phase speed solve the ODE
b u - ~u + (~ - 1) u = 0 xx 2 x 2 (B.5)
where b is a small parameter multiplying the dispersive term. This transforms to
If we expand the solution as a perturbation series in b, i. e.,
00
u = L bj u(j)
j=O
then the lowest order solution is a polynomial in z:
uO = 1- z
However, logarithms appear at higher order:
u(l) = -2(1 - z2) log(l - z2)
(B.7)
(B.8)
(B.9)
U(2) = 4(1 - z2) {z log2(1 - Z2) + 2z log(l - z2) + 6z + 2 log (~ ~ ;) }
(B.IO)
488 ApPENDIX B
Nevertheless, the transformation may still be useful in coaxing a solution from the differential equation: both manual analysis and symbolic manipulation languages are happier with polynomial coefficients than with powers of hyperbolic functions.
c ELLIPTIC FUNCTIONS
"The theory of elliptic functions is the fairyland of mathematics. The mathematician who once gazes upon this enchanting and wondrous domain crowded with the most beautiful relations and concepts is forever captivated."
- Richard E. Bellman
C.l BASIC PROPERTIES
If
(C.l)
where x=sin(</» (C.2)
then am(u; k) == </> (C.3)
sn(u; k) == sin(</» = x (C.4)
cn(u;k) == cos(</» = ~ (C.5)
489
490 ApPENDIX C
Values at u = 0:
cn(O; k) = 1, sn(O; k) = 0, dn(O; k) = 1, am(O; k) = 0 (C.6)
Symmetries:
cn( -u) = cn(u),
dn( -u) = dn(u),
sn( -u) = - sn(u)
am(-u) = -am(u)
Connections:
cn2 + sn2
dn2 + k2 sn2
dn2 - k2 cn2
C.2 ELLIPTIC NOME AND MODULUS
(C.7)
(C.8)
(C.g)
(C.lD)
(C.lI)
(C.12)
The complete elliptic integral of the first kind is usually denoted by K(k).
There are five interchangeable parameters to specify the degree of ellipticity of an elliptic function. Specifying anyone of these five parameters immediately and uniquely determines the other four. The five members of this set are
1. k "elliptic modulus"
2. m = k2 , also confusingly called the "elliptic modulus"
3. k' = v'f="k2 "complementary modulus"
4. q = exp( -7rK( v'f="k2)/ K(k))
5. S = K(k)/K(v'f="k2), which has no standard name and is related to the nome via
q = exp( -7r / S) (C.13)
Elliptic Functions
C.3 SERIES: COMPLETE ELLIPTIC INTEGRAL AND MODULUS
K is the elliptic integral of the first kind.
491
Note: nome q = exp(-7r/S). These converge for all S but converge must rapidly for either S > 1 or S < 1 as indicated.
K(S)
k(S)
=
00
= Si + S7r I.: sech (j7r S) j=l
k is the elliptic modulus.
L~-oo sech«j + 1/2)7r/S)
L~-oo sech (j7r / S)
L~-oo (-l)j sech (j7rS)
L~-oo sech (j7rS)
[S < 1]
[S> 1]
[8 < 1]
[S> 1]
Alternative series for the modulus:
k(S) = L~-oo (-l)jcosech«(j + 1/2)7r/S)
[S < 1] L~-oo sech (j7r / S)
= L~-oo (-l)j tanh «1/2 - j) 7rS)
[S> 1] L~-oo sech (j 7rS)
(C.14)
(C.15)
(C.16)
(C.17)
(C.18)
(C.19)
492 ApPENDIX C
C.4 ELLIPTIC FUNCTIONS: FOURIER & IMBRICATE SERIES
Note: sn and cn are periodic with period 4K in u and 21r in y. The function dn is periodic with period 2K in u and 1r in y.
sn
cn
=
2K 1r u=-Yf-'ty=-u
1r 2K
21r 00 qn-l/2 . ( 1r) kK L 1 _ 2n-l SIll (2n - 1) 2 K u
n=l q
Z=:=-oo cosech((n+1/2)1r/S) exp(i(2n+1)y)
i Z=:=-oo (-1)ncosech((n+1/2)1r/S)
Z=:=-oo (-l)m tanh(S(y - 1rm))
Z=:=-oo (-l)m sech(m1rS)
21r 00 qn-l/2 ( 1r) kK L 1+q2n-l cos (2n-1)2K u
n=l Z=:=-oo sech((n+1/2)1r/S) exp(i(2n+1)y)
Z=:=-oo sech((n+1/2)1r/S)
Z=:=-oo (-l)m sech(S(y - 1rm))
Z=:=-oo (-l)m sech(m1rS)
= 1r 2 1r ~ qn (1r) dn 2K + K ~ 1 + q2n cos 2n 2 K u
n=l
= Z=:=-oo sech (n1r / S) exp( i2ny)
Z=:=-oo sech (n1r/S)
Z=:=-oo sech(S(y - 1rm))
Z=:=-oo sech(m1rS)
(C.20)
(C.21)
(C.22)
(C.23)
(C.24)
(C.25)
(C.26)
(C.27)
(C.28)
(C.29)
Elliptic Functions 493
C.5 ELLIPTIC FUNCTIONS: DIFFERENTIATION IDENTITIES
d du sn(u) = en dn (C.30)
d du cn(u) = - sn dn (C.31)
d du dn(u) = _k2 sn en (C.32)
Second Derivatives:
d2
du2sn(u) = -(1+k2)sn + 2k2sn3 (C.33)
d2 du2 cn(u) = (2e - 1) en - 2k2 cn3 (C.34)
d2 du2dn(u) = (2 - e) dn - 2dn3 (C.35)
Notational convention: In the variable u, sn and en are periodic with period 4K whereas dn is periodic with period 2K where K is the complete elliptic integral of the first kind.
C.6 ELLIPTIC FUNCTIONS: INTEGRATION
J sn(u)du = ~log{dn - kcn(u)}
J cn(u) du = ~ arccos { dn(u)}
J dn(u) du = arcsin { sn(u) }
(C.36)
(C.37)
(C.38)
D SOLITONS AND CNOIDAL WAVES:
KDV, MKDV, NLS MIXED KDV/MKDV
"The theory of inverse scattering for the KdV equation is so beautiful that it helps me to forget the loss of much of my pension."
- Chia-Shun Yih (1918-1997), who was teaching himself KdV theory during the stock market crash of 1987.
D.l SOLITONS
D.1.1 KdV
The stationary (that is, the independent-of-time ordinary differential equation) form of the KdV equation is
(D.1)
This has the solitary wave
(D.2)
Solitary waves are possible regardless of the sign of the quadratic coefficient b. However, the solitons have the property
sign(u) = sign(b) (D.3)
For the KdV application of D.l, the phase speed is the negative of the coefficient of the linear term, that is,
(D.4)
494
Solitons and Cnoidal Waves 495
D.1.2 MKdV /NLS
The stationary form of the Modified Korteweg-de Vries (MKdV) equation is the same as the stationary form of the Nonlinear Schrodinger equation (NLS):
(D.5)
Its soliton is
U = ± VI fsech(fx) (D.6)
Restriction: solitons are possible only if b > O. If this is satisfied, then MKdV solitary waves of both elevation and depression exist. For the MKdV application of D.5, the phasE' <,;oeed is the negative of the coefficient of the linear term, that is,
(D.7)
For the NLS equation, the sech function is the shape of the envelope of the wave packet.
D.1.3 KdV /MKdV with Mixed Cubic-and-Quadratic N onlinearity
The mixed KdV /MKdV equation is
U xx - 4102 U + f u2 + 9 u3 = 0
Its soliton is
u 6f2~+ 1 1 f cosh2(fX) - ~ sinh2(fx)
6f2~+1 1 f ~ + (1 -~) COSh2(fX)
(D.8)
(D.9)
where the cubic coefficient of the differential equation is given in terms of the solitary wave and the quadratic coefficient by
(D.lO)
or equivalently, the soliton parameter ~ is a root of
(D.ll)
496 ApPENDIX D
Restriction: JL E [-00,1) since the soliton has a pole for real x when JL 2: 1. Properties:
1. u decays monotonically with Ixl for all values of parameters.
2. sign(u) = sign {(I + JL)/J} for all values of JL,f,j,g.
3. When 9 < 0, f2 ~ P /(18g).
4. In terms of the parameter
(D.12)
the roots for JL are (D.13)
5. When 8 is small, JL ~ -1 ±..j'jl and u ~ ±y'87gfsech(2fx), which is the same as MKdV soliton except for the rescaling f --> 2f.
6. JL is real-valued only when 8 is not on the range 8 E [-2,0] or equivalently, JL is real only if -g < j2/(18f2).
References: Kakutani and Yamasaki(1978) and Miles(1979).
D.1.4 Benjamin-Davis-Ono
The equation, with amplitude and length scaled so as to give unit coefficients, is
-C:Ux + uUx - H(uxx ) = 0
Denoting the phase speed by c, the soliton is
1 u = -4c 2 2' C > 0
l+cx
(D.14)
(D.15)
D.2 SPATIALLY PERIODIC SOLUTIONS
D.2.1 KdV
Note that this section employs the third-order rather than second order form of the KdV equation. The spatial period is P.
uxxx - C:Ux + uUx = 0 (D.16)
Solitons and Cnoidal Waves
00
u(x) M - 24; + 121:2 .L sech2 (I:(x - Pj)) )=-00
4rr2 00 n M + 12 p2 L. (mr2) cos([2rr/P]nx)
j=1 smh €p
497
(D.17)
where M and I: > 0 are arbitrary constants. Note that M is merely an additive shift added simultaneously to both u and c. The coefficient of cos (x) is
a =
The inverse relationship is
4rr2 12
p2 sinh (;;)
4rr2 24 _ q_ p2 1- q2
1 I: P arcsinh ( 48rr2 / ( aP2)) ,
The elliptic nome is
P ~ 48 a
The phase speed is given by
a» 1
c = u(x) + ~: {~ [n4/sinh(:;)] sin([~] nx)} /
{t, [~n2;mnh (:;) 1 ffin ([~ 1 nx) } 00
M - 24; + 41:2 - 241:2 L cosech 2 ( n PI:) n=1
(D.18)
(D.19)
(D.20)
(D.21)
The first line is derived from the differential equation, that is, c = u + uxxx/u, and gives the same value for c for any x. The second line is from Whitham(1984), rescaled to a different period.
Stokes' series: expansion in powers of a, the amplitude of cos([2rr / Px). Define the auxiliary parameters
p 2
v=-2' 4rr 2rr
K,=--P (D.22)
498 ApPENDIX D
(D.24)
D.2.2 MKdV
The MKdV /NLS boundary value problem is
Uxx - cu +2u3 =0. (D.25)
This has two classes of solutions. For both, c = (uxx + 2u3 )/u independent of x, which is sometimes a convenient alternative to the elliptic integral formulas below.
DNOIDAL BRANCH: The dnoidal wave is the elliptic function dn. For weak waves, this
branch asymptotes to a constant.
00
u(x) = ±f L sech(f(x-Pj)) (D.26) j=-oo
(D.27)
(D.28)
where the parameter S = ('Tr/P)f, and the elliptic integral K(S) and modulus k( S) and the elliptic function dn can be calculated by the formulas of Appendix C, Secs. 3 and 4.
c
(D.29)
Solitons and Cnoidal Waves 499
CNOIDAL BRANCH: The cnoidal branch is approximately a cosine function for small am-
plitude.
00
u(X) € L (-l)i sech (€(x - jP/2)) (D.30) i=-oo
4kK (4K ) -----p- cn p Xj k (D.32)
where the parameter S = (2 7r / P)€ [different from the ratio for dn], and the elliptic integral K(S) and modulus k(S) and the elliptic function en can be calculated by the series of Appendix C, Secs. 3 and 4.
Note that c becomes negative for small amplitude waves. (In contrast, the solitary wave speed is always positive.)
(D.33)
D.2.3 Benjamin-Davis-Ono
The equation, with amplitude and length scaled so as to give unit coefficients, is
-cux + uUx - H(uxx ) = 0 (D.34)
For general period P and in terms of the positive parameter €,
27r (27r) C = pcotanh P€ (D.35)
u 27r tanh(~:) - 2 - ---...,..,,--,,--'-':"">:":_--
P 1- sech (~:) cos(27rx/P)
- 2 ~ {1+ 2 ~ exp( -2~n/['P]) cos(2~nx/ P) }
00
-4 " ~ 1 + €2(X - np)2 n=-oo
(D.36)
E TIME INTEGRATION WITH THE
FOURIER PSEUDOSPECTRAL ALGORITHM FOR WAVE
EQUATIONS
"Mighty are numbers, and joined with art irresistable." - Euripides, Hecuba, line 884.
E.l SPECTRAL METHODS AND WAVE EQUATIONS
Solitary wave problems are usually posed in idealized geometry. An engineer at Boeing may compute the flow around a complicated, spinning turbine inside an engine, but for the FKdV, TNLS and other simple wave equations, the domain is usually spatially periodic - sometimes for physical reasons and sometimes as an approximation to an unbounded domain. The sines and cosines of a Fourier series are the optimum spectral basis for a periodic interval, and the only basis discussed in this appendix.
Some authors have obtained good results using finite difference methods. However, the Fourier pseudospectral basis is very easy and cheap to implement. The exponential accuracy of a spectral method is highly desirable when looking for exponentially small effects in nonlocal solitary waves.
E.2 TIME INTEGRATION SCHEMES
When a time-dependent partial differential equation is discretized in the spatial coordinates, the result is a system of ordinary differential equations (ODEs) in time of the form
dil ~ dt = F(il, t) (E.1)
500
Time Integration 8 Fourier Pseudospectral Algorithm 501
Table E.! Time-Marching with a Fourier Basis
References Comments Boyd(1989a, 1998c) Monograph on spectral methods Canuto et al.(1987) Monograph on spectral methods
Fornberg&Sloan (1994) Review; comparisons of pseudospectral & finite difference Fornberg(1996) Monograph on pseudospectral algorithms
Fornberg& Whitham( 1978) Nonlinear waves.: KdV, MKdV, Benjamin-Ono & others Garcia-Archilla(1996) 'Equal Width' equation
Herbst&Ablowitz (1992) Sine-Gordon eqn.; numerical instabilities; integrable-to-chaos transition because of numerical errors
Herbst&Ablowitz (1993) Symplectic time-marching, numerical chaos, exponentially small splitting of separatrices
If&Berg&Christiansen Split-step spectral for Nonlinear Schrodinger Eq. & Skovgaard(1987) with absorbing (damping) boundary conditions
Mulholland&Sloan(1992) Implicit & semi-implicit with preconditioning for wave equations
Sanders&Katopodes KdV, RLW, Boussinesq eqs. &Boyd (1997)
Weideman&James (1992) Benjamin-Ono equation Tan&Boyd(1997) Two-dimensional generalization of quasi-geostrophic eq. Boyd&Tan(1998) Solitary vortices, topographic deformations
Vallis(1985) Doubly-periodic quasi-geostrophic flow
where it(t) is the vector containing the unknowns. In the pseudospectral algorithm, as in a conventional finite difference spatial discretization, the elements of it are the values of u(x, t) at the points of a discrete grid in x. The generic form (E.l) applies whether the governing equations are a single equation or a system, whether there is one space coordinate or two or three, whether the equations are linear or nonlinear, and lastly whether the spatial discretization is spectral, finite difference, or finite element.
The Fourier pseudospectral method is a particularly natural choice of spatial discretization for wave equations with periodic boundary conditions. Two tasks remain. The first is to efficiently evaluate the vector-valued function F which is the right-hand side of the system of ODEs in time. This will be explained in the next section.
The other task is to advance the ODE system in time. Since Eq.(E.l) is in standard form, any good book of algorithms will furnish a choice of methods. Press et al.(1986) provides software listings and C, Fortran, or Pascal (depending on the edition).
502 ApPENDIX E
After careful analysis of both dispersion errors and dissipation errors for this and a variety of other popular algorithms, Durran (1991) rates the third AdamsBashforth scheme (AB3) above all others:
un+1 = un + T {~F (un, x, tn) - ~ F (un-l, x, t n - 1) } (E.2)
where n denotes the time level. AB3 requires only a single evaluation of F per time step.
Runge-Kutta (RK) methods are much more costly per step than AB3. However, compared to the pure fourth order scheme it is only slightly more expensive to simultaneously evaluate the fifth order formula, compare the two approximations, and adaptively vary the time step so as to stay within a user-set error tolerance. Furthermore, the stability limit of RK4/5 is roughly three times that of AB3. In addition, the adaptive RK subroutines do not require the user to specify a timestep a priori, but can internally calculate a timestep which is both stable and accurate. Press et al.(1986) gives software listings; Matlab has a built-in routine called ode45.
When one can estimate a good timestep in advance, the AB3 method is cheaper. However, the greater robustness and accuracy of adaptive RK4/5 has made it popular, too.
E.3 PSEUDOSPECTRAL EVALUATION OF THE RIGHT-HAND SIDE: THE FAST FOURIER TRANSFORM
The key to efficiently computing the right-hand side of the ODE system is: Multiply in physical space and differentiate in spectral space. To explain this principle, a little background is necessary.
The unknown u(x) is approximated by a truncated Fourier series
N/2-1
u(x) ~ L ak exp(ikx) k=-N/2
(E.3)
For expository simplicity, we assume the spatial period is x E [-IT, ITJ. If the period is actually y E [-P/2, P/2J, one can apply the method as outlined here
Time Integration f3 Fourier Pseudospectral Algorithm 503
by making the trivial change of coordinate
x == (21r/ P) Y (E.4)
The elements Uj of the vector il are the set of values of u(x) at the points of evenly spaced grid
j=I,2, ... ,N (E.5)
The "interpolation points" or "collocation points" are
Xj = 1r {-I + 2(j -1)/N}, j=I,2, ... ,N (E.6)
The grid points values {Uj} and the Fourier coefficients {ak} are equivalent approximations of u(x). They are connected by summation (a --t u) and interpolation (u --t a). Both these transforms can performed in roughly (5/2)NdZ092 (N) operations by using the Fast Fourier Transform in d space dimensions.
Differentiation is trivial in spectral space: If the coefficients of the j-th derivative are denoted by a(j), then
(E.7)
The indefinite integral, as needed for the RMKdV equation, is just a special case of the same formula:
(E.8)
The RMKdV solution is constrained to have zero mean, that is, ao = 0 (Chap. 15), so the apparent singularity at k = 0 is an illusion. The coefficients of the Hilbert Transform, as needed for the Benjamin-Ono equation, are given by
at: = isgn(k)ak
where at: denotes the coefficients of 1i(u).
(E.9)
The time-integration cycle to evaluate the right-hand side F( il, t) of the system of ODEs in time then consists of the following (Fig. E.l):
1. Compute the spectral coefficients ak of u( x) by a forward Fourier Transform.
2. Compute the coefficients of the derivatives a~) (and similarly those of the Hilbert Transform and integral if needed) by applying (E.7), (E.8) and (E.9).
504 ApPENDIX E
3. Apply the inverse Fourier Transform to compute the grid point values of the derivatives, etc., of u(x) from the corresponding spectral coefficients.
4. Evaluate nonlinear terms and variable coefficient terms by multiplication of grid point values, i. e., the grid point values of uUx are the products of Uj with ux(Xj).
The cost per evaluation, in any number of space dimensions, is proportional to the number of grid points multiplied by 10g(N). On a modern workstation, even multi-space-dimensional equations can be solved in little time.
Table E.2 shows the remarkable simplicity of the Fourier pseudospectral algorithm for the KdV equation where the elements of Fare -u(Xj, t)ux(Xj, t) -uxxx(Xj, t). It calls "fIt" and "ifIt", which are built-in MATLAB Fast Fourier Transform routines. It employs MATLAB's operation for elementwise multiplication of one vector by another vector, which is denoted by". *". A FORTRAN subroutine would be a little longer because of the need to replace the elementwise multiplications by DO loops and so on. Nevertheless, the brevity of the subroutine is startling.
The first line computes the Fourier coefficients, a vector a, by taking a Fourier Transform. The second line computes the coefficients of the first and third
.... ~ ............. .
Fast Fourier Transform
L~::~~!L .. ; <:Il <:Il \J \J
'= '= ~
;::.. Cl)
"- "-'= '=
........... ) ~ Uj ~ ..........
\J ... VI ;::-,
...:: ~
~ \J <:Il ;::..
Cl)
\ \ Fast Fourier "-I Transform
xik
Figure E.1 Schematic of the calculation of the righthand side of the system of ordinary differential equations in time that results from Fourier pseudospectral spatial discretization. The calculation of each term (schematically represented here by the single term -uux) flows from a representation in terms of the grid point values ("Physical space") to Fourier coefficients ak ("Spectral Space") and then back again. The gateway from one representation to another is the Fast Fourier Transform. Derivatives are calculated in spectral space (right side); multiplication is performed using the grid point value representation (left side).
Time Integration &J Fourier Pseudospectral Algorithm 505
Table E.2 MATLAB code for Right-Hand Side of the ODE System
function F=KdVRHS(t,u)j global k kcub a=ifft(u)j % Compute Fourier coefficients an from grid point values u(Xj) ax=k . * aj axxx = kcub . * aj % Compute coefficients of 1st and 3rd derivative ux=real(fft(ax))j uxxx=real(fft(axxx) )j % Reverse FFT to get grid point values
% of first and third derivatives F= - u . * ux - UXXXj % RHS of KdV ODE system. Nonlinear term evaluated by
%pointwise multiplication of U by U x
% In a preprocessing step, either in the main program or in a subroutine called once, % one must execute the following to initialize the vectors k and kcub with the product % of i with the wavenumber k and with the (negative) of its cube, respectively for j=1:(n/2), k(j)=-i*(j-1)j endj for j=(n/2+1):n, k(j)=-i*(j-1 - n)j end for j=1:n, kcub(j)=k(j)*k(j)*k(j), end
derivatives by multiplying those of u(x) by (ik) and (-ik3 ), respectively. (It is assumed that in a preprocessing step, the vectors "k" and "kcub" have been initialized with the appropriate wavenumbers). The third step is to take inverse Fourier Transforms to convert these coefficients for the derivatives into the corresponding grid point values. The final step is to add the grid point values together to form the vector F. Note that the nonlinear term is evaluated by point-by-point multiplication of the grid point values of u with those of U x .
E.4 COMPLICATION I: LIBRARY FAST FOURIER TRANSFORM SOFTWARE
One virtue of the Fast Fourier Transform is that one never needs to write code to do it; all software libraries have well-optimized subroutines. Unfortunately, most compute a sum of wavenumbers from k = 0 to N - 1 (or k = 1 to N) instead of k = - N /2 to k = N /2 - 1, which is the usual convention in solving differential equations. One must read the software documentation very carefully and then modify the user-written code so that high wavenumbers (k > N/2) are properly interpreted as negative wavenumbers. (Note that wavenumbers k and k - N are indistinguishable on a grid of N points; each is the "alias" of the other on the discrete grid.)
506 ApPENDIX E
Be warned! In twenty years of teaching and consulting, this seems to be the most common practical difficulty in writing pseudospectral codes.
E.5 COMPLICATION 11: DIFFERENTIAL OPERATORS ACTING ON TIME DERIVATIVES AS IN THE REGULARIZED LONG WAVE AND QUASI-GEOSTROPHIC EQUATIONS
Some partial differential equations have time derivatives multiplied by a differential operator L, i. e.,
L Ut = G(u,x,t)
To restate this in the canonical form
Ut = F(u,x,t),
we must solve a boundary value for F:
LF=G
(E.1O)
(E.Il)
(E.12)
For such an equation, it is as laborious to apply an explicit method as an implicit method because one must solve a boundary value problem at every time step. Such equations are called "implicitly-implicit" because the boundary-valuesolving labor of an implicit time-marching scheme cannot be avoided even when applying an explicit algorithm like AB3.
Two implicitly-implicit examples are the Regularized Long Wave (RLW) equation of water wave theory,
(E.13)
and the quasi-geostrophic equation of meteorology and physical oceanography,
(E.14)
Happily, when the boundary conditions are spatial periodicity, the inversion of the operator L is trivial for both examples.
For the RLW equation, for example, the first step is to evaluate
(E.15)
Time Integration fj Fourier Pseudospectral Algorithm 507
The second step is compute the Fourier coefficients gk by an FFT. In spectral space, the boundary value problem Eq.(E.12) is
(E.16)
where the {Id are the spectral coefficients of F(u, x, t) and where we have the used the fact that the second derivative of exp(ikx) is _k2 exp(ikx). An inverse FFT then gives the grid point values of F.
The RLW equation was invented because the differential operator acting on Ut
drastically slows the phase speed of short waves, allowing a much longer time step than for KdV without computational instability. The quasi-geostrophic equation was invented for similar reasons and used for the first computer weather forecasts in 1950. It continued in use until 1965 when the first CDC6600 made it possible to forecast with the "primitive equations" .
However, modern semi-implicit algorithms use as long a time step for the primitive equations as for the quasi-geostrophic equation. The semi-implicit boundary value problem is just as cheap to solve as the quasi-geostrophic problem. The difference is that the primitive equations generate more accurate forecasts because this system does not filter out low frequency gravity waves and Kelvin waves, as quasi-geostrophy does.
Similarly, the KdV equation has been exorcised of its former frightfulness by a combination of fast workstations, which can solve one-dimensional problems quickly even with a tiny time step, and semi-implicit time-marching algorithms, which allow it to be integrated as quickly as its understudy, the RLW equation. (Canuto et al., 1987, and Boyd, 1989a, 1998c). Smart algorithms have largely replaced clever approximation.
Nevertheless, if one still wants to solve an "implicitly-implicit' wave equation, the Fourier pseudospectral method will work just fine.
GLOSSARY
"Pure mathematics is about what is provable. Applied mathematics is about what is."
- J. P. Boyd (1990)
ALGEBRAIC CONVERGENCE: If the error in a series decreases asO(N-k )
where k has a finite upper bound, then the series has algebraic convergence.
ARITHMURGY: Synonym for "number-crunching". (From the Greek ap(}Ji,Oa,
"number", and -f(Y'foa, ''working''.)
ASYMPTOTIC: A power series is asymptotic to a function !(f) if, for fixed N and sufficiently small f, I!(f) - Ef=o aj fjl « fN (Bender and Orszag, 1978).
ASYMPTOTIC RATE OF CONVERGENCE: IfaserieshasGEOMETRlC CONVERGENCE, that is, if
lanl < exp( -nJ1.) all n,
then the "asymptotic rate of convergence" is the largest J1. for which the above bound is true. The asymptotic rate of convergence is zero for series with algebraic or subgeometric convergence.
BANDED (MATRIX): A label applied to a matrix whose elements Aij are all zero except for diagonal bands around the main diagonal.
BANDWIDTH (of a MATRIX) If the elements Aij = 0 unless li-jl ::; m, then m is the BANDWIDTH of the BANDED MATRlX.
508
Glossary 509
BASIS RECOMBINATION: A strategy for satisfying numerical boundary conditions in which the original basis set, such as Chebyshev polynomials, is replaced by a new basis composed of linear combinations of the original basis functions such that each member of the new basis individually satisfies the boundary conditions. The alternative strategy is BOUNDARY BORDERING.
BASIS FUNCTIONS: The members of a basis set. Examples of basis functions are the Chebyshev polynomials and the Hermite functions.
BASIS SET: The collection of functions which are used to approximate the solution of a differential equation. The Fourier functions {1, cos(nx) , sin(nx) for n = 1,2, ... } and the Chebyshev polynomials {Tn(x), n = 0, 1, ... } are two examples of basis sets.
BEHAVIORAL BOUNDARY CONDITION: A boundary condition that imposes a certain behavior on the solution rather specifying a numerical constraint. Examples of behavioral boundary conditions include: (i) periodicity with a certain period L (ii) boundedness and infinite differentiability at a point where the coefficients of the differential equation are singular. It is usually possible to satisfy such conditions by proper choice of basis function. For example, the sines and cosines of a Fourier series are periodic, so the terms of a Fourier basis individually satisfy the behavioral boundary condition of periodicity.
BELL SOLITON: A solitary wave which has a sech-like or "bell" -like shape such as the soliton of the KdV Eq.
BEYOND-ALL-ORDERS PERTURBATION THEORY: Catch-all term for any and all perturbative methods that calculate terms which are exponentially small in 1/f. where f. is the perturbation parameter. Such terms are smaller than f.n for any finite value of n as f. --+ 0, and therefore cannot be calculated by a power series expansion in f.. [Synonym: HYPERASYMPTOTIC.]
BIFURCATION POINT: A point where two branches of a solution u(a) cross. Synonym is "crossing point". For both a > a bif and a > a bif,
two different solutions exist which meet at the bifurcation point. Newton's method fails at a bifurcation point, but it is possible to "shoot the bifurcation point" or switch branches (Chapter 8). (When "bifurcation" is used in a broader sense, a "crossing point" is also called a "trans-critical bifurcation" .)
510 GLOSSARY
BOUNDARY BORDERING: A strategy for satisfying boundary conditions in which these are imposed as explicit constraints in the matrix problem which is solved for the spectral coefficients. Some of the collocation or Galerkin conditions on the residual are replaced by a "border" of rows which impose the constraints. The major alternative is BASIS RECOMBINATION.
BREATHER: A class of solitary wave in which the coherent structure has a standing wave oscillation superimposed upon its translational motion, if any. The function u(x, t) = sech(Ex) sin(wt) has the form of a breather.
CHOLESKY FACTORIZATION: Factorization of a SYMMETRIC matrix into the product LLT where L is lower triangular and LT is its transpose; Cholesky factorization requires only half as many operations as the usual L U factorization.
CNOIDAL WAVE: A spatially-periodic generalization of a solitary wave. The term was coined by Korteweg and deVries, who showed that the KdV equation had exact solutions that could be expressed in terms of the elliptic cosine function, whose abbreviation is "cn". ("-oid" is Greek for "like", so "cnoidal" means "cn-like".) By extension, this term is now applied to similar steadily-translating, spatially periodic solutions, regardless of whether these are described by the "cn" function or not.
COLLOCATION: Adjective for labelling methods which determine the series coefficients by demanding that the residual function be zero on a grid of points. In this book, a synonym for PSEUDOSPECTRAL and for METHOD OF SELECTED POINTS. Elsewhere in the literature, this adjective is also applied to certain finite element methods.
COLLOCATION POINTS: the grid of points where the residual function must vanish in a pseudospectral algorithm. Synonym for INTERPOLATION POINTS.
COMPATIBILITY CONDITIONS: A countably infinite set of constraints on the initial conditions of a time-dependent partial differential which are necessary for the solution to be analytic everywhere in the space-time domain. Example: an incompressible flow will be strongly singular at t = 0 if the initial condition is divergent at some point in the fluid, and will have discontinuities (vortex sheets) at rigid walls if the initial velocity does not satisfy the no-slip boundary condition of vanishing at the walls. The flow will be singular, but more weakly in the sense of having bounded derivatives of higher order, if the initial flow does not satisfy additional constraints. The constraints are different for each partial differential equation.
Glossary 511
CORE of a soliton: The region around the point where the solitary wave has its maximum or minimum value, and the nonlinear terms in the differential equation describing the soliton are very important.
CROUT REDUCTION: Solution of a matrix equation by direct LU FACTORlZATION.
DARBOUX'S PRINCIPLE: Theorem that the asymptotic form of the spectral coefficients an as n ----+ 00 for a function f (x) are controlled by the singularities (poles, branch points, etc.) of f(x).
DIPOLE VORTEX: A pair of vortices rotating in opposite directions, forming a coherent, bound state.
DIRECT MATRIX METHOD: A non-iterative algorithm for solving a matrix equation such as Gaussian elimination, Crout reduction, Cholesky factorization, LU factorization, skyline solver, and static condensation.
DISCRETE ORDINATES METHOD: Synonym for "PSEUDOSPECTRAL" or "COLLOCATION" method. (This term is most common among physicists and chemists.)
DISCRETIZATION ERROR: This is the difference between the first (N + 1) coefficients of the exact solution and the corresponding coefficients as computed using a spectral or pseudospectral algorithm using (N + 1) basis functions. It is distinct from (but usually the same order-of-magnitude as) the TRUNCATION ERROR.
DNOIDAL WAVE: A spatially-periodic generalization of a solitary wave which can be expressed in terms of the elliptic "dn". (Usually applied to one of the two families of periodic solutions of the Nonlinear Schrodinger equation.)
DOMAIN TRUNCATION: A method of solving problems on unbounded domains by replacing the interval yE [-00, 00] by yE [-L, L]. If lu(±L)1 decays exponentially with L, then it is possible to obtain solutions of arbitrary accuracy by choosing L sufficiently large.
DOUBLE CNOIDAL WAVE: The 2-POLYCNOIDAL wave, that is, a nonlinear wave which is independent of time except for steady translation due to two independent phase variables. In other words, u(x, t) = u(X, Y) where X == kl(x - Clt) and Y = kl(x - Clt).
ENVELOPE OF THE COEFFICIENTS: A smooth, monotonically decreasing curve which is a tight bound on oscillatory Chebyshev or Fourier
512 GLOSSARY
coefficients {an} in the sense that the absolute value of the coefficients is arbitrarily close to the envelope infinitely often as n --t 00. (Borrowed from wave theory, where the "envelope of a wave packet" has the identical meaning.)
ENVELOPE SOLITARY WAVE: A nonlinear wave packet which is the product of a spatially localized function ("envelope") that propagates at the group velocity cg together with a sinusoidal factor ("carrier wave") which travels at the phase velocity cp. A function of the form sech(x -cgt) sin( k[x - cpt]) has the structure of an envelope solitary wave where the sech function is the envelope and the sine function is the carrier wave.
EQUAL ERRORS, ASSUMPTION OF : This empirical principle, unprovable but supported by strong heuristic arguments and practical experience, states that the "DISCRETIZATION ERROR" and "TRUNCATION ERROR" and "INTERPOLATION ERROR" are the same order of magnitude.
EXPLICIT: A time-integration scheme in which the solution at the next time level is given by an explicit formula that does not require solving a boundary value problem.
EXPONENTIAL CONVERGENCE: A spectral series possesses the property of "exponential convergence" if the error decreases faster than any finite inverse power of N as N, the number of terms in the truncated series, increases. Typically, the series coefficients decrease as O(exp[-pnrJ) for some positive constants p and r, which is the reason for the adjective "exponential". A synonym for "infinite order convergence" .
FAR FIELD of a soliton: Region far from the core of the solitary wave where the coherent structure has decayed to such small amplitude that the nonlinear terms in the equation of motion can be neglected.
FFT: Abbreviation for FAST FOURIER TRANSFORM.
FKdV Abbreviation for the Fifth-Order Korteweg-deVries equation Ut+uux+
uxxx+uxxxxx =0.
FOLD POINT: Synonym for LIMIT POINT.
GEOMETRIC CONVERGENCE: A series whose coefficients decrease
an rv A(n)pn t-7 A(n) exp[-n I log(p) ID n --t 00 Ipl < 1
has the property of "geometric convergence" where A( n) denotes a function that varies algebraically with n (such as a power of n). The reason for the
Glossary 513
name is that the terms of a geometrically convergent series can always be bounded by those of a geometric series, that is, by the terms of the power series expansion of a/(f3 + x) for some a and f3 where these are constants. [All convergent power series have geometric convergence. All Chebyshev series for functions which have no singularities on x E [-1, 1] (including the endpoints) also have geometric convergence.]
HOMOCLINIC: A solution to an ordinary differential equation (ODE) which returns to the same fixed point. A classical solitary wave is an example of a homo clinic trajectory (in terms of the moving coordinate, X == x - et, because as X increases from -00, the solitary wave rises from zero, reaches a maximum, and then returns to zero again. "Homoclinic" is more general because it applies to the solutions of ODEs in time as well as to solitons.
HOMOCLINIC-to-PERIODIC-ORBIT: A solution to an ordinary differential equation (ODE) which returns to the same periodic trajectory (limit cycle). A weakly nonlocal solitary wave is homo clinic to a periodic orbit.
HYPERASYMPTOTIC: A form of BEYOND-ALL-ORDERS perturbation theory in which the error is reduced below the SUPERASYMPTOTIC error by appending one or more terms of a second asymptotic expansion with different scaling assumptions than those of the primary series. Loosely applied to all BEYOND-ALL-ORDERS methods.
IMBRICATE SERIES A representation of a spatially periodic function which is the superposition of an infinite number of evenly spaced identical copies of a "pattern" function A(x). All periodic functions have imbricate series in addition to their Fourier expansions, and often the imbricate series converge faster. Imbricate series may be generalized to an arbitrary number of dimensions.
IMPLICIT: A time-integration scheme in which it is necessary to solve a boundary-value problem to compute the solution at the new time level. The Crank-Nicholson method is an example.
INFINITE ORDER CONVERGENCE: A spectral series possesses the property of "infinite order convergence" if the error decreases faster than any finite inverse power of N as N, the number of terms in the truncated series, increases. A synonym for "exponential convergence".
INTERPOLANT: An approximation PN-l(X) whose free parameters or coefficients are chosen by the requirement that
i = 1, ... , N
514 GLOSSARY
at a set of N grid points. The process of computing such an approximation is INTERPOLATION.
INVARIANT TORUS: A multiply-periodic solution to a system of differential equations. The N-torus is an N-dimensional surface in the phase space of the system which is periodic in all N coordinates. An N-POLYCNOIDAL wave is an example of an invariant torus. The surface is "invariant" in the sense that solution remains always on the toroidal surface, and therefore remains always multiply-periodic. (Terminology of dynamical systems theory.)
KdV Abbreviation for the Korteweg-deVries equation Ut + UUx + U xxx = O.
KG Abbreviation for the Klein-Gordon equation. The Cubic KG equation is U xx - Utt ± (u - u3 ) = O.
KYMOLOGY: The study of waves. (From the Greek "IWJ-La", ''wave'').
LIMIT POINT: A point where a solution u( a) of a nonlinear equation curves back so that there are two solutions for a on one side of a = a limit and no solutions on the other side of the limit point. As the limit point is approached, du/da -> 00. Special methods ["pseudoarclength continuation" or "globally convergent homotopy"j are needed to "turn the corner" and march from the lower branch through the limit point onto the upper branch or vice versa. Synonyms are "fold point", ''turning point", and "saddle-node bifurcation" .
LOCALIZED, SPATIALLY: A label applied to a wave or wave disturbance whose amplitude decreases rapidly as Ixl -> 00.
MATCHED ASYMPTOTICS, COMPLEX PLANE: A variant of the singular perturbation method known as matched asymptotics in which the matching is performed at points in the complex X-plane to calculate a radiation coefficient or reflection coefficient a which is exponentially small for real X.
MICROPTERON: A weakly nonlocal solitary wave whose far field wings are an algebmic, non-exponential function of the width of the core. (From the Greek J-LU'i,poa, "small", and 7rTfpOV, "wing".)
MKdV Abbreviation for the Modified Korteweg-deVries equation Ut + UUx + Uxxx = o.
MMT: MATRIX MULTIPLICATION TRANSFORM This is N-point interpolation using N-point Gaussian quadrature or summation of an Npoint spectral series at each of the N grid points. In either direction, the
Glossary 515
transform is computed by multiplying an N -dimensional column vector by aN-dimensional square matrix.
MOD ON A DIPOLE VORTEX, composed of two contra-rotating vortices of equal strength, which steadily translates as a coherent structure Also known as "Batchelor dipoles" or more accurately, as "Lamb-Chapyglin Vortex Pairs".
MONOPOLE VORTEX A spinning cylindrical column of fluid in which the vorticity [curl of velocity] is one-signed in the vortex core.
NANOPTERON: A weakly nonlocal solitary wave whose far field wings are an exponentially small function of the width of the core. (From the Greek I/QI/oa, "small", and 1fTEPOI/, "wing".)
NLS Abbreviation for the Nonlinear Schroedinger equation i At + (1/2)Axx ± A\A\2 = o.
NONLOCAL, SPATIALLY: A steadily translating disturbance which fills all of space or a wave which is slowly decaying in time through radiation which will eventually fill all space.
NUMERICAL BOUNDARY CONDITION: A constraint such as u( -1) = -0.5 which involves a number. It is always necessary to modify either the basis set or pseudospectral matrix to enforce such conditions.
OPTIMALLY-TRUNCATED ASYMPTOTIC SERIES: If an asymptotic series is divergent, then for a given E, the error decreases as more terms are added up to some Nopt (E) and then increases. The "optimal truncation" is to include only those terms up to and including O(ENopt(e»). An optimally-truncated series is said to be a SUPERASYMPTOTIC approximation.
PERIODIC: A function f(x) is "periodic" with period L if and only if
f(x + L) = f(x)
for all x.
POLYCNOIDAL WAVE: A spatially-periodic generalization of a multiple soliton solution. The N-polycnoidal wave can be written exactly as a function of N phase variables of the form Xj == x - Cjt + cPj, i = 1,2, ... N where the Cj are phase speeds and the cPj are phase constants. In the language of dynamical systems theory, polycnoidal waves are INVARIANT TOR!.
516 GLOSSARY
PRECONDITIONING: A technique for accelerating the convergence of an iteration for solving Ax - f by changing the iteration matrix to H- 1 A. The matrix H is the "preconditioning matrix" and is chosen to approximate A (in the sense of having approximately the same eigenvalues), but is also constructed to be much less expensive to invert than A.
PSEUDOSPECTRAL: an algorithm which uses an interpolation grid to determine the coefficients of a spectral series. Synonyms are ORTHOGONAL COLLOCATION, METHOD OF SELECTED POINTS & METHOD of DISCRETE ORDINATES.
RADIATION COEFFICIENT: a. The amplitude of the far field oscillations.
RESIDUAL FUNCTION: When an approximate solution UN is substituted into a differential, integral, or matrix equation, the result is the RESIDUAL function, usually denoted R(x; aa, ab ... , aN)' The residual function would be identically zero if the approximate solution were exact.
RESURGENCE: Literally, the "act of rising again". In hyperasymptotics, a principle used to postpone the rise in the terms of a divergent ordinary asymptotic series to higher degree (and lower error). (A neologism of J. Ecalle (1981).)
RMKdV: Rotation-Modified Korteweg-de Vries equation, 8 x (Ut + UUx + uxxx )
f 2U = O. (Also called the "Ostrovsky" equation.)
RMKP: Rotation-Modified Kadomtsev-Petviashvili equation, Ox (Ut + UUx + uxxx )-
102 (u - U yy ) = O.
RULE OF THREE NAMES: Every term in this glossary has at least two synonyms.
SEMI-IMPLICIT: a time-integration method that treats some terms implicitly and others explicitly. Such algorithms are very common in hydrodynamics.
SOLITARY WAVE (CLASSICAL): A steadily-translating, finite amplitude wave that decays to zero as one moves away from the core of the disturbance.
SOLITARY WAVE (NONLOCAL): A nonlinear wave which satisfies the definition of a classical solitary wave except for the leakage of radiation to infinity. If the wave is spatially localized, then the core is accompanied by ever-spreading oscillatory wings of radiation. If the wave is of permanent
Glossary 517
form, then the core is accompanied by small amplitude standing oscillations that extend to infinity. The far field radiation cannot be suppressed by any small perturbation of the shape of the disturbance.
SOLITON: A synonym for "SOLITARY WAVE". Some authors restrict this term to solitary waves which collide elastically and solve integrable equations amendable to the inverse scattering method, but we shall not.
SOLITON, RADIATIVELY DECAYING: A nonlocal solitary wave which is spatially localized but decays with time through radiation to spatial infinity.
SPECTRAL: A catch-all term for all methods (including pseudospectral techiques) which expand the unknown as a series of global, infinitely differentiable expansion functions.
SUB GEOMETRIC CONVERGENCE: Two equivalent definitions. The convergence rate of the series is "subgeometric" if (i) The series converges exponentially fast with n, but too slowly for the coefficients to be bounded in magnitude by c exp( -pn) for any positive constants c and p, or equivalently, if (ii)
lim log(lani)/n = o. n-+oo
All known examples are the expansions of functions which are singular but infinitely differentiable at some point (or points) on the expansion interval (often at the endpoints). In mathematical jargon, such weakly singular functions "are in Coo , but not en."
SUPERASYMPTOTIC: A label for an OPTIMALLY-TRUNCATED asymptotic series. This term is convenient because the error in an optimallytruncated expansion is typically O(exp( -q/f)) for some constant q even though the individual terms of the series are proportional to powers of the perturbation parameter f.
SUPERIOR LIMIT: For a sequence {an}, the superior limit or supremum limit is written lim sup{ an} and denotes the lower bound of the almost upper bounds of the sequence. (A number is an almost upper bound for a sequence if only a finite number of members of the sequence exceed the "almost upper bound" in value.) Strictly speaking, definitions of convergence rates should be expressed in terms of superior limits, rather than ordinary limits, to allow for oscillations and zeros in the sequence as n ---t 00. A synonym is "supremum limit" .
518 GLOSSARY
SYMMETRIC: (i) [Of a matrix]: Aij = A ji . The longer but more precise term "CENTROSYMMETRlC" is sometimes used as a synonym. (ii) [Of a function f(x)]: f(x) = f( -x) for all x.
TENSOR PRODUCT BASIS: A multi-dimensional basis whose elements are the products of one-dimensional basis elements. In two dimensions,
TENSOR PRODUCT GRID: A multi-dimensional grid whose M N points are chosen from the corresponding one-dimensional grids:
i = 1, ... , M & j = 1, ... , N
TNLS Abbreviation for the Third-Order Nonlinear Schrodinger equation i At + (1/2)Axx ± AIAI2 - i Axxx = o.
TRUNCATION ERROR: the error made by neglecting all coefficients an in the spectral series such that n > N for some "truncation" N.
WEAKLY NONLOCAL SOLITARY WAVE: A steadily-translating, finite amplitude wave that decays to a small amplitude oscillation (rather to zero) as one moves away from the core of the disturbance.
WINGS of a nanopteron: The small amplitude oscillations in the far field of a weakly nonlocal solitary wave.
cp4 FIELD THEORY This is the partial differential equation CPxx - CPtt + (cpcp3) = 0 , or equivalently, with u == cp - 1, Uxx - Utt - 2u - 3u2 - u3 = O.
REFERENCES
Abdullah, J.: 1955, The atmospheric solitary wave, Bull. Amer. Meteor. Soc. 10, 511-518.
Ablowitz, M. J. and Segur, H.: 1981, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, Pennsylvania. Good treatment of classical, exactly integrable solitary waves but nothing about nonlocal solitary waves.
Abramowitz, M. and Stegun, I. A.: 1965, Handbook of Mathematical Functions, Dover, New York.
Ackerberg, R C. and O'Malley, Jr., RE.: 1970, Boundary layer problems exhibiting resonance, Stud. Appl. Math. 49, 277-295. Classical paper illustrating the failure of standard matched asymptotics; this can be resolved by incorporating exponentially small terms in the analysis (MacGillivray, 1997).
Acton, F.: 1970, Numerical Methods As They Should Be, Harper and Row, New York.
Adamson, Jr., T. C. and Richey, G. K.: 1973, Unsteady transonic flows with shock waves in two-dimensional channels, J. Fluid Mech. 60, 363-382. Show the key role of exponentially small terms.
Afimov, G. L., Eleonsky, V. M., Kulagin, N. E., Lerman, L. M. and Silin, V. P.: 1989, On the existence of nontrivial solutions for the equation .6.u-u+u3 = 0, Physica D 44,168-177. Numerical calculation of classical soliton.
Airey, J. R: 1937, The "converging factor" in asymptotic series and the calculation of Bessel, Laguerre and other functions, Philosophical Magazine 24, 521-552. Hyperasymptotic approximation to some special functions for large 1xJ.
Akylas, T. R: 1991, On the radiation damping of a solitary wave in a rotating channel, in T. Miloh (ed.), Mathematical Approaches in Hydrodynamics, SIAM, Philadelphia, Pennsylvania, pp. 175-181.
519
520 WEAKLY NONLOCAL SOLITARY WAVES
Akylas, T. R: 1994, Three-dimensional long water-wave phenomena, Ann. Rev. Fluid Mech. 26, 191. Review including the RMKP and RMKdV equations, which have weakly nonlocal solitons.
Akylas, T. Rand Grimshaw, R H. J.: 1992, Solitary internal waves with oscillatory tails, J. Fluid Mech. 242,279-298. Theory agrees with observations of Farmer and Smith (1980).
Akylas, T. Rand Kung, T.-J.: 1990, On nonlinear wave envelopes of permanent form near a caustic, J. Fluid Mech. 214, 489-502. TNLS.
Akylas, T. Rand Yang, T.-S.: 1995, On short-scale oscillatory tails of longwave disturbances, Stud. Appl. Math. 94, 1-20. Nonlocal solitary waves; perturbation theory in Fourier space.
Alvarez, G.: 1988, Coupling-constant behavior of the resonances of the cubic anharmonic oscillator, Phys. Rev. A 37, 4079-4083. Beyond-ail-orders perturbation theory in quantum mechanics.
Amick, C. J. and Kirchgassner, K.: 1989, Solitary water-waves in the presence of surface tension, Archive Rat. Mech. Anal. 105,1-49. Derive fourth order ODE (FKdV) equation for water waves through center manifold theory; prove existence of classical soliton for Bond number> 1/3.
Amick, C. J. and McLeod, J. B.: 1990, A singular perturbation problem in needle crystals, Archive Rat. Mech. Anal. 109, 139-171.
Amick, C. J. and McLeod, J. B.: 1991, A singular perturbation problem in water waves, Stability and Applied Analysis in Continuous Media 1, 127-148. Proof of the nonexistence of classical solitons for the FKdV equation.
Amick, C. J. and Toland, J.: 1992, Solitary waves with surface tension I: Trajectories homoclinic to periodic orbits in four dimensions, Archive Rat. Mech. Anal. 118, 37-69. Existence of weakly nonlocal solitons for the FKdV equation; estimates that the radiation coefficient is smaller than any finite power of 102 , the amplitude of the core.
Amick, C. J., Ching, E. S. C., Kadanoff, L. P. and Rom-Kedar, v.: 1992, Beyond all orders: Singular perturbations in a mapping, J. Nonlinear Sci. 2,9-67.
Arnold, V. I.: 1978, Mathematical Methods of Classical Mechanics, SpringerVerlag, New York. Quote about why series diverge: pg.395.
References 521
Arteca, G. A., Fernandez, F. M. and Castro, E. A.: 1990, Large Order Perturbation Theory and Summation Methods in Quantum Mechanics, SpringerVerlag, New York. 642pp.; beyond-aIl-orders perturbation theory.
Baer, F.: 1977, Adjustment of initial conditions required to suppress gravity oscillations in non-linear flows, Contributions to Atmospheric Physics 50, 350-366. Divergent multiple scales series to approximate the slow manifold.
Baer, F. and Tribbia, J. J.: 1977, On complete filtering of gravity modes through non linear initialization, Mon. Weather Rev. 105, 1536-1539. Divergent multiple scales series to approximate the slow manifold.
Baker, Jr., G. A.: 1965, Pade approximants, Advances in Theoretical Physics 1, Academic Press, New York, pp. 1-50.
Baker, Jr., G. A.: 1975, Essentials of Pade Approximants, Academic Press, New York. 220 pp.
Balian, R., Parisi, G. and Voros, A.: 1978, Discrepancies from asymptotic series and their relation to complex classical trajectories, Phys. Rev. Letters 41(17),1141-1144. Quantum beyond-aIl-orders perturbation theory.
Balian, R., Parisi, G. and Voros, A.: 1979, Quartic oscillator, in S. Albeverio, P. Combe, R. Hoegh-Krohn, G. Rideau, M. Siruge-Collin, M. Sirugue and R. Stora (eds) , Feynman Path Integrals, number 106 in Lecture Notes in Physics, Springer-Verlag, New York, pp. 337-360.
Bank, R. E. and Chan, T. F.: 1986, PLTMGC: A multi-grid continuation program for parameterized nonlinear elliptic systems, SIAM J. Sci. Stat. Comput. 7, 540--559.
Beale, J. T.: 1991a, Exact solitary water waves with capillary ripples at infinity, Comm. Pure Appl. Math. 44, 211-257.
Beale, J. T.: 1991b, Solitary water waves with ripples beyond all orders, in H. Segur, S. Tanveer and H. Levine (eds), Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 293-298.
Bekyarov, K. L. and Christov, C. I.: 1991, Fourier-Galerkin numerical technique for solitary waves of fifth order Korteweg-DeVries equation, Chaos, Solitons and Praetals 5, 423-430. Rational function spectral method on the infinite interval.
522 WEAKLY NONLOCAL SOLITARY WAVES
Ben Amar, M. and Combescot, R.: 1991, Saffman-Taylor viscous fingering in a wedge, in H. Segur, S. Tanveer and H. Levine (eds), Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 155-173.
Benassi, L., Grecchi, V., Harrell, E. and Simon, B.: 1979, Bender-Wu formula and the Stark effect in hydrogen, Phys. Rev. Letters 42, 704-707. Exponentially small corrections in quantum mechanics.
Bender, C. M. and Orszag, S. A.: 1978, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York. 594 pp.
Benilov, E., Grimshaw, R. H. and Kuznetsova, E.: 1993, The generation ofradiating waves in a singularly perturbed Korteweg-deVries equation, Physica D 69,270-276.
Benjamin, T. B.: 1967, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29, 559-592. BDO Eq. derivation.
Benney, D. J.: 1966, Long non-linear waves in fluid flows, Journal of Mathematics and Physics 45, 52-63. Rossby solitons.
Berestov, A. L.: 1981, Some new solutions for Rossby solitons, Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 17,60-64. Three-dimensional classical modons.
Berestov, A. L. and Monin, A. S.: 1980, Solitary Rossby waves, Adv. Mech. 3, 3-34. Review.
Berman, A. S.: 1953, Laminar flow in channel with porous walls, J. Appl. Phys. 24, 1232-1235. Earliest paper on an ODE (Berman-Robinson problem) where exponentially small corrections are important.
Berry, M. V.: 1989a, Stokes' phenomenon; smoothing a Victorian discontinuity, Publicationes Mathematiques IHES 68, 211-221.
Berry, M. V.: 1989b, Uniform asymptotic smoothing of Stokes's discontinuities, Proc. Roy. Soc. London A 422, 7-21.
Berry, M. V.: 1990a, Waves near Stokes lines, Proc. Roy. Soc. London A 427, 265-280.
Berry, M. V.: 1990b, Histories of adiabatic quantum transitions, Proc. Roy. Soc. London A 429, 61-72.
Berry, M. V.: 1991a, Infinitely many Stokes smoothings in the Gamma function, Proc. Roy. Soc. London A 434, 465-472.
References 523
Berry, M. V.: 1991b, Stokes phenomenon for superfactorial asymptotic series, Proc. Roy. Soc. London A 435, 437-444.
Berry, M. V.: 1991c, Asymptotics, superasymptotics, hyperasymptotics, in H. Segur, S. Tanveer and H. Levine (eds), Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 1-14.
Berry, M. V.: 1994a, Faster than Fourier, in J. S. Auandan and J. L. Safko (eds) , Quantum Coherence and Reality: in Celebmtion of the 60th Birthday of Yakir Aharonov, World Scientific, Singapore.
Berry, M. V.: 1994b, Evanescent and real waves in quantum billiards and Gaussian beams, J. Phys. A 27, L391-L398.
Berry, M. V.: 1994c, Asymptotics, singularities and the reduction of theories, in D. Prawitz, B. Skyrms and D. Westerstahl (eds) , Logic, Methodology and Philosophy of Science IX, Elsevier, Amsterdam, pp. 597-607.
Berry, M. V.: 1995, Riemann-Siegel expansion for the zeta function: high orders and remainders, Proc. Roy. Soc. London A 450, 439-462. Beyond all orders asymptotics.
Berry, M. V. and Howls, C. J.: 1990a, Hyperasymptotics, Proc. Roy. Soc. London A 430, 653-668.
Berry, M. V. and Howls, C. J.: 1990b, Stokes surfaces of diffraction catastrophes with codimension three, Nonlinearity 3, 281-291.
Berry, M. V. and Howls, C. J.: 1990c, Fake Airy functions and the asymptotics of refiectionlessness, J. Phys. A 23, L243-L246.
Berry, M. V. and Howls, C. J.: 1991, Hyperasymptotics for integrals with saddles, Proc. Roy. Soc. London A 434, 657-675.
Berry, M. V. and Howls, C. J.: 1993a, Unfolding the high orders of asymptotic expansions with coalescing saddles: Singularity theory, crossover and duality, Proc. Roy. Soc. London A 443, 107-126.
Berry, M. V. and Howls, C. J.: 1993b, Infinity interpreted, Physics World pp. 35-39.
Berry, M. V. and Howls, C. J.: 1994a, Overlapping Stokes smoothings: survival of the error function and canonical catastrophe integrals, Proc. Roy. Soc. London A 444, 201-216.
524 WEAKLY NONLOCAL SOLITARY WAVES
Berry, M. V. and Howls, C. J.: 1994b, High orders of the Weyl expansion for quantum billiards: Resurgence of periodic orbits, and the Stokes phenomenon, Proc. Ray. Soc. London A 447, 527-555.
Berry, M. V. and Keating, J. P.: 1992, A new approximation for ((1/2+it) and quantum spectral determinant, Proc. Ray. Soc. London A 437,151-173.
Berry, M. V. and Lim, R.: 1993, Universal transition prefactors derived by superadiabatic renormalization, J. Phys. A 26, 4737-4747.
Bhattacharyya, K.: 1981, Notes on polynomial perturbation problems, Chem. Phys. Lett. 80, 257-261.
Bhattacharyya, K. and Bhattacharyya, S. P.: 1980, The sign-change argument revisited, Chem. Phys. Lett. 76, 117-119. Criterion for divergence of asymptotic series.
Bhattacharyya, K. and Bhattacharyya, S. P.: 1981, Reply to "another attack on the sign-change argument", Chem. Phys. Lett. 80, 604-605.
Bohe, A.: 1990, Free layers in a singularly perturbed boundary value problem, SIAM J. Math. Anal. 21, 1264-1280.
Bokhove, O. and Shepherd, T. G.: 1996, On Hamiltonian balanced dynamics and the slowest invariant manifold, J. Atmos. Sci. 53, 276-297. Apply KAM theory and Poincare sections to the inviscid Lorenz-Krishnamurthy Quintet with heavy emphasis on chaos.
Boyd, J. P.: 1976, The noninteraction of waves with the zonally averaged flow on a spherical earth and the interrelationships of eddy fluxes of heat, energy, and momentum, J. Atmos. Sci. 33, 2285-229l.
Boyd, J. P.: 1978a, A Chebyshev polynomial method for computing analytic solutions to eigenvalue problems with application to the anharmonic oscillator, J. Math. Phys. 19, 1445-1456.
Boyd, J. P.: 1978b, The choice of spectral functions on a sphere for boundary and eigenvalue problems: A comparison of Chebyshev, Fourier and associated Legendre expansions, Mon. Weather Rev. 106, 1184-1191.
Boyd, J. P.: 1978c, Spectral and pseudospectral methods for eigenvalue and nonseparable boundary value problems, Mon. Weather Rev. 106, 1192-1203.
Boyd, J. P.: 1978d, The effects of latitudinal shear on equatorial waves, Part 1: Theory and methods, J. Atmos. Sci. 35, 2236-2258.
References 525
Boyd, J. P.: 1978e, The effects of latitudinal shear on equatorial waves, Part 11: Applications to the atmosphere, J. Atmos. Sci. 35, 2259-2267.
Boyd, J. P.: 1980a, The nonlinear equatorial Kelvin wave, J. Phys. Oceangr. 10, 1-11.
Boyd, J. P.: 1980b, The rate of convergence of Hermite function series, Math. Comp. 35, 1309-1316.
Boyd, J. P.: 1980c, Equatorial solitary waves, Part I: Rossby solitons, J. Phys. Oceangr. 10, 1699-1718.
Boyd, J. P.: 1981a, A Sturm-Liouville eigenproblem with an interior pole, J. Math. Phys. 22, 1575-1590.
Boyd, J. P.: 1981b, The rate of convergence of Chebyshev polynomials for functions which have asymptotic power series about one endpoint, Math. Comp. 37, 189-196.
Boyd, J. P.: 1981c, Analytical approximations to the modon dispersion relation, Dyn. Atmos. Oceans 6, 97-101.
Boyd, J. P.: 1982a, The optimization of convergence for Chebyshev polynomial methods in an unbounded domain, J. Comput. Phys. 45, 43-79. Infinite and semi-infinite intervals; guidelines for choosing the map parameter or domain size L.
Boyd, J. P.: 1982b, The effects of meridional shear on planetary waves, Part I: Nonsingular profiles, J. Atmos. Sci. 39, 756-769.
Boyd, J. P.: 1982c, The effects of meridional shear on planetary waves, Part 11: Critical latitudes, J. Atmos. Sci. 39, 770-790. First application of cubicplus-linear mapping with spectral methods. The detour procedure of Boyd (1985a) is better in this context.
Boyd, J. P.: 1982d, A Chebyshev polynomial rate-of-convergence theorem for Stieltjes functions, Math. Comp. 39, 201-206.
Boyd, J. P.: 1982e, Theta functions, Gaussian series, and spatially periodic solutions of the Korteweg-de Vries equation, J. Math. Phys. 23, 375-387.
Boyd, J. P.: 1983a, Equatorial solitary waves, Part 11: Envelope solitons, J. Phys. Oceangr. 13, 428-449.
Boyd, J. P.: 1983b, Long wave/short wave resonance in equatorial waves, J. Phys. Oceangr. 13, 450-458.
526 WEAKLY NONLOCAL SOLITARY WAVES
Boyd, J. P.: 1983c, Second harmonic resonance for equatorial waves, J. Phys. Oceangr. 13, 459-466.
Boyd, J. P.: 1983d, The continuous spectrum of linear Couette flow with the beta effect, J. Atmos. Sci. 40, 2304-2308.
Boyd, J. P.: 1984a, The asymptotic coefficients of Hermite series, J. Comput. Phys. 54, 382-410.
Boyd, J. P.: 1984b, Equatorial solitary waves, Part IV: Kelvin solitons in a shear flow, Dyn. Atmos. Oceans 8, 173-184.
Boyd, J. P.: 1984c, Cnoidal waves as exact sums of repeated solitary waves: New series for elliptic functions, SIAM J. Appl. Math. 44, 952-955. Imbricate series for nonlinear waves.
Boyd, J. P.: 1984d, The double cnoidal wave of the Korteweg-de Vries equation: An overview, J. Math. Phys. 25, 3390-3401.
Boyd, J. P.: 1984e, Perturbation theory for the double cnoidal wave of the Korteweg-de Vries equation, J. Math. Phys. 25, 3402-3414.
Boyd, J. P.: 1984f, The special modular transformation for the polycnoidal waves of the Korteweg-de Vries equation, J. Math. Phys. 25, 3390-3401.
Boyd, J. P.: 1985a, Complex coordinate methods for hydrodynamic instabilities and Sturm-Liouville problems with an interior singularity, J. Comput. Phys. 57, 454-471.
Boyd, J. P.: 1985b, Equatorial solitary waves, Part 3: Modons, J. Phys. Oceangr. 15, 46-54.
Boyd, J. P.: 1985c, An analytical and numerical study of the two-dimensional Bratu equation, J. Sci. Comput. 1, 183-206. Nonlinear eigenvalue problem with 8-fold symmetry.
Boyd, J. P.: 1985d, Barotropic equatorial waves: The non-uniformity of the equatorial beta-plane, J. Atmos. Sci. 42, 1965-1967.
Boyd, J. P.: 1986a, Solitons from sine waves: analytical and numerical methods for non-integrable solitary and cnoidal waves, Physica D 21,227-246. Fourier pseudospectral with continuation and the Newton-Kantorovich iteration.
Boyd, J. P.: 1986b, Polynomial series versus sinc expansions for functions with corner or endpoint singularities, J. Comput. Phys. 64, 266-269.
References 527
Boyd, J. P.: 1987a, Exponentially convergent Fourier/Chebyshev quadrature schemes on bounded and infinite intervals, J. Sci. Comput. 2, 99-109.
Boyd, J. P.: 1987b, Spectral methods using rational basis functions on an infinite interval, J. Comput. Phys. 69, 112-142.
Boyd, J. P.: 1987c, Orthogonal rational functions on a semi-infinite interval, J. Comput. Phys. 70, 63-88.
Boyd, J. P.: 1987d, Generalized solitary and cnoidal waves, in G. Brantstator, J. J. Tribbia and R. Madden (eds), NCAR Colloquium on Low Frequency Variability in the Atmosphere, National Center for Atmospheric Research, Boulder, Colorado, pp. 717-722. Numerical calculation of the exponentially small wings of the 4;4 breather.
Boyd, J. P.: 1988a, Chebyshev domain truncation is inferior to Fourier domain truncation for solving problems on an infinite interval, J. Sci. Comput. 3, 109-120.
Boyd, J. P.: 1988b, An analytical solution for a nonlinear differential equation with logarithmic decay, Adv. Appl. Math. 9, 358-363. df/dt = exp(-l/f), which models radiative decay of nanopterons.
Boyd, J. P.: 1988c, The superiority of Fourier domain truncation to Chebyshev domain truncation for solving problems on an infinite interval, J. Sci. Comput. 3, 109-120.
Boyd, J. P.: 1989a, Chebyshev and Fourier Spectral Methods, Springer-Verlag, New York. 792 pp.
Boyd, J. P.: 1989b, New directions in solitons and nonlinear periodic waves: Polycnoidal waves, imbricated solitons, weakly non-local solitary waves and numerical boundary value algorithms, in T.-y' Wu and J. W. Hutchinson (eds), Advances in Applied Mechanics, number 27 in Advances in Applied Mechanics, Academic Press, New York, pp. 1-82.
Boyd, J. P.: 1989c, Periodic solutions generated by superposition of solitary waves for the quarticly nonlinear Korteweg-de Vries equation, ZAMP 40, 940-944. Imbrication of solitary wave generates good approximate periodic solutions.
Boyd, J. P.: 1989d, The asymptotic Chebyshev coefficients for functions with logarithmic endpoint singularities, Appl. Math. Comput. 29,49-67.
528 WEAKLY NONLOCAL SOLITARY WAVES
Boyd, J. P.: 198ge, Non-local equatorial solitary waves, in J. C. J. Nihoul and B. M. Jamart (eds), Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence: Proc. 20th Liege Coll. on Hydrodynamics, Elsevier, Amsterdam, pp. 103-112. Typo: In (4.1b), 0.8266 should be 1.6532.
Boyd, J. P.: 1990a, The orthogonal rational functions of Higgins and Christov and Chebyshev polynomials, J. Approx. Theory 61, 98-103.
Boyd, J. P.: 1990b, A numerical calculation of a weakly non-local solitary wave: the 4>4 breather, Nonlinearity 3, 177-195. The eigenfunction calculation (5.15, etc.) has some typographical errors corrected in Chapter 12.
Boyd, J. P.: 1990c, The envelope of the error for Chebyshev and Fourier interpolation, J. Sci. Comput. 5,311-363.
Boyd, J. P.: 1990d, A Chebyshev/radiation function pseudospectral method for wave scattering, Computers in Physics 4,83-85. Numerical calculation of exponentially small reflection.
Boyd, J. P.: 1991a, A comparison of numerical and analytical methods for the reduced wave equation with multiple spatial scales, App. Numer. Math. 7,453-479. Study of uxx ±ux = f(Ex). Typo: E2n factor should be omitted from Eq. (4.3).
Boyd, J. P.: 1991b, Monopolar and dipolar vortex solitons in two space dimensions, Wave Motion 57, 223-243.
Boyd, J. P.: 1991c, Nonlinear equatorial waves, in A. R. Osborne (ed.), Nonlinear Topics of Ocean Physics: Fermi Summer School, Course LIX, NorthHolland, Amsterdam, pp. 51-97.
Boyd, J. P.: 1991d, Weakly nonlocal solitary waves, in A. R. Osborne (ed.), Nonlinear Topics of Ocean Physics: Fermi Summer School, Course LIX, North-Holland, Amsterdam, pp. 527-556.
Boyd, J. P.: 1991e, Weakly non-local solitons for capillary-gravity waves: Fifthdegree Korteweg-de Vries equation, Physica D 48, 129-146. Typo: at the beginning of Sec. 5, 'Newton-Kantorovich (5.1)' should read 'NewtonKantorovich (3.2)'. Also, in the caption to Fig. 12, '500,000' should be '70,000'.
Boyd, J. P.: 1991f, Sum-accelerated pseudospectral methods: The Euleraccelerated sinc algorithm, App. Numer. Math. 7, 287-296.
References 529
Boyd, J. P.: 1992a, The arctan/tan and Kepler-Burger mappings for periodic solutions with a shock, front, or internal boundary layer, J. Comput. Phys. 98,181-193. Numerical trick which is useful for solitary waves and cnoidal waves.
Boyd, J. P.: 1992b, The energy spectrum of fronts: The time evolution of shocks in Burgers' equation, J. Atmos. Sei. 49, 128-139.
Boyd, J. P.: 1992c, Multipole expansions and pseudospectral cardinal functions: A new generalization of the Fast Fourier Transform, J. Comput. Phys. 102, 184-186.
Boyd, J. P.: 1992d, A fast algorithm for Chebyshev and Fourier interpolation onto an irregular grid, J. Comput. Phys. 103, 243-257.
Boyd, J. P.: 1992e, Defeating the Runge phenomenon for equispaced polynomial interpolation via Tikhonov regularization, Appl. Math. Lett. 5,57-59.
Boyd, J. P.: 1993a, Chebyshev and Legendre spectral methods in algebraic manipulation languages, J. Symb. Comput. 16, 377-399.
Boyd, J. P.: 1994a, Hyperviscous shock layers and diffusion zones: Monotonicity, spectral viscosity, and pseudospectral methods for high order differential equations, J. Sei. Comput. 9, 81-106.
Boyd, J. P.: 1994b, The rate of convergence of Fourier coefficients for entire functions of infinite order with application to the Weideman-Cloot sinh-mapping for pseudospectral computations on an infinite interval, J. Comput. Phys. 110, 360-372.
Boyd, J. P.: 1994c, The slow manifold of a five mode model, J. Atmos. Sci. 51, 1057-1064.
Boyd, J. P.: 1994d, Nonlocal modons on the beta-plane, Geophys. Astrophys. Fluid Dyn. 75, 163-182.
Boyd, J. P.: 1994e, Time-marching on the slow manifold: The relationship between the nonlinear Galerkin method and implicit timestepping algorithms, Appl. Math. Lett. 7,95-99.
Boyd, J. P.: 1994f, Sum-accelerated pseudospectral methods: Finite differences and sech-weighted differences, Comput. Meth. Appl. Meeh. Engr. 116, Ill.
Boyd, J. P.: 1995a, Weakly nonlocal envelope solitary waves: Numerical calculations for the Klein-Gordon (4)4) equation, Wave Motion 21,311-330.
530 WEAKLY NONLOCAL SOLITARY WAVES
Boyd, J. P.: 1995b, A hyperasymptotic perturbative method for computing the radiation coefficient for weakly nonlocal solitary waves, J. Comput. Phys. 120, 15-32.
Boyd, J. P.: 1995c, Eight definitions of the slow manifold: Seiches, pseudoseiches and exponential smallness, Dyn. Atmos. Oceans 22, 49-75.
Boyd, J. P.: 1995d, A lag-averaged generalization of Euler's method for accelerating series, Appl. Math. Comput. 72, 146-166.
Boyd, J. P.: 1995e, A Chebyshev polynomial interval-searching method ("Lanczos economization") for solving a nonlinear equation with application to the nonlinear eigenvalue problem, J. Comput. Phys. 118, 1-8.
Boyd, J. P.: 1995f, Multiple precision pseudospectral computations of the radiation coefficient for weakly nonlocal solitary waves: Fifth-Order KortewegdeVries equation, Computers in Physics 9, 324-334.
Boyd, J. P.: 1996a, Asymptotic Chebyshev coefficients for two functions with very rapidly or very slowly divergent power series about one endpoint, Appl. Math. Lett. 9(2), 11-15.
Boyd, J. P.: 1996b, Traps and snares in eigenvalue calculations with application to pseudospectral computations of ocean tides in a basin bounded by meridians, J. Comput. Phys. 126, 11-20.
Boyd, J. P.: 1996c, Numerical computations of a nearly singular nonlinear equation: Weakly nonlocal bound states of solitons for the Fifth-Order Korteweg-deVries equation, J. Comput. Phys. 124, 55-70.
Boyd, J. P.: 1996d, The Erfc-Log filter and the asymptotics of the Vandeven and Euler sequence accelerations, in A. V. Hin and L. R. Scott (eds), Proceedings of the Third International Conference on Spectral and High Order Methods, Houston Journal of Mathematics, Houston, Texas, pp. 267-276.
Boyd, J. P.: 1997a, Pade approximant algorithm for solving nonlinear ODE boundary value problems on an unbounded domain, Computers and Physics 11(3), 299-303. FP monopole is an example.
Boyd, J. P.: 1997b, The periodic generalization of Camassa-Holm "peakons": An exact superposition of solitary waves, Appl. Math. Comput. 81(2), 173-187. Classical solitons.
Boyd, J. P.: 1997c, Construction of Lighthill's unitary functions: The imbricate series of unity, Appl. Math. Comput. 86(1), 1-10.
References 531
Boyd, J. P.: 1998a, Radiative decay of weakly nonlocal solitary waves, Wave Motion. In press.
Boyd, J. P.: 1998b, Pseudospectral/Delves-Freeman computations of the radiation coefficient for weakly nonlocal solitary waves of the Third Order Nonlinear Schroedinger Equation and their relation to hyperasymptotic perturbation theory, J. Comput. Phys. Submitted.
Boyd, J. P.: 1998c, Chebyshev and Fourier Spectral Methods, Dover, New York. Second edition of Boyd(1989a), to appear.
Boyd, J. P.: 1998d, High order models for the nonlinear shallow water wave equations on the equatorial beta-plane fwith application to kelvin wave frontogenesis, Dyn. Atmos. Oceans. Submitted.
Boyd, J. P.: 1998e, The Devil's Invention: Asymptotics, superasymptotics and hyperasymptotics, Acta Applicandae. Submitted.
Boyd, J. P.: 1998f, Two comments on filtering, J. Comput. Phys. Submitted.
Boyd, J. P. and Chen, G.-Y.: 1998, Analytical and numerical studies of weakly nonlocal solitary waves of the Rotation-Modified Korteweg-deVries equation. In preparation.
Boyd, J. P. and Christidis, Z. D.: 1982, Low wavenumber instability on the equatorial beta-plane, Geophys. Res. Lett. 9, 769-772. Growth rate is exponentially in 1/€ where € is the shear strength.
Boyd, J. P. and Christidis, Z. D.: 1983, Instability on the equatorial betaplane, inJ. Nihoul (ed.), Hydrodynamics of the Equatorial Ocean, Elsevier, Amsterdam, pp. 339-351.
Boyd, J. P. and Christidis, Z. D.: 1987, The continuous spectrum of equatorial Rossby waves in a shear flow, Dyn. Atmos. Oceans 11, 139-151.
Boyd, J. P. and Haupt, S. E.: 1991, Polycnoidal waves: Spatially periodic generalizations of multiple solitary waves, in A. R. Osborne (ed.), Nonlinear Topics of Ocean Physics: Fermi Summer School, Course LIX, NorthHolland, Amsterdam, pp. 827-856.
Boyd, J. P. and Ma, H.: 1990, Numerical study of elliptical modons by a spectral method, J. Fluid Meeh. 221, 597-611.
Boyd, J. P. and Moore, D. W.: 1986, Summability methods for Hermite functions, Dyn. Atmos. Oceans 10, 51-62. Numerical.
532 WEAKLY NONLOCAL SOLITARY WAVES
Boyd, J. P. and Tan, B.: 1998, Vortex crystals and non-existence of nonaxisymmetric solitary waves in the Flierl-Petviashvili equation, Physica D. Submitted. Double Fourier algorithm for a generalized, two-dimensional quasi-geostrophic equation. Classical solitons.
Boyd, W. G. C.: 1990e, Stieltjes transforms and the Stokes phenomenon, Proc. Roy. Soc. London A 429, 227-246.
Boyd, W. G. C.: 1993b, Error bounds for the method of steepest descents, Proc. Roy. Soc. London A 440, 493-516.
Boyd, W. G. C.: 1993c, Gamma function asymptotics by an extension of the method of steepest descents, Proc. Roy. Soc. London A 447, 609-630.
Boyd, W. G. C.: 1996e, Steepest-descent integral representations for dominant solutions of linear second-order differential equations, Methods and Applications of Analysis 3(2), 174-202.
Branis, S. V., Martin, O. and Birman, J. L.: 1991, Self-induced transparency selects discrete velocities for solitary-wave solutions., Phys. Rev. A 43, 1549-1563. Nonlocal envelope solitons.
Brazel, N., Lawless, F. and Wood, A. D.: 1992, Exponential asymptotics for an eigenvalue of a problem involving parabolic cylinder functions, Proc. Amer. Math. Soc. 114,1025-1032.
Buffoni, B., Champneys, A. R. and Toland, J. F.: 1996, Bifurcation and coalescence of a plethora of multi-modal homoclinic orbits in a Hamiltonian system, J. Diff. Eq. 8, 221-281. Nothing about nonlocal solitons, but a very complete analysis of the classical solitary waves of the FKdV equation including extensive numerics and a proof that there are infinitely many classical bions.
Bulakh, B. M.: 1964, On higher approximations in the boundary-layer theory, J. Appl. Math. Mech. 28, 675-681.
Buryak, A. V.: 1995, Stationary soliton bound states existing in resonance with linear waves, Phys. Rev. E 52, 1156-1163. TNLS nonlocal bions and (for discrete amplitudes) classical TNLS bions.
Buryak, A. V. and Akhmediev, N. N.: 1995, Phys. Rev. E 51, 1156. TNLS nonlocal bions and (for discrete amplitudes) classical TNLS bions.
Buryak, A. V. and Champneys, A. R.: 1997, On the stability of solitary wave solutions of the 5th-order KdV equation, Phys. Lett. Submitted. Confirmed that classical FKdV bions with decaying oscillatory tails, which have two
References 533
large peaks at one of an infinite number of quantized separations, are alternately stable and unstable as the separation distance is increased from one allowed separation distance to the next for fixed amplitude of the core peaks. Unstable bions fission into two ordinary solitary waves, each with a single core peak.
Burzlaff, J. and Wood, A. D.: 1991, Optical tunneling from a one-dimensional square-well potential, IMA J. Appl. Math. 47, 207-215.
Byatt-Smith, J. G. and Davie, A. M.: 1990, Exponentially small oscillations in the solution of an ordinary differential equation, Proc. Royal Soc. Edinburgh A 114, 243-.
Byatt-Smith, J. G. and Davie, A. M.: 1991, Exponentially small oscillations in the solution of an ordinary differential equation, in H. Segur, S. Tanveer and H. Levine (eds) , Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 223-240.
Calvo, D. C. and Akylas, T. R: 1997a, The formation of bound states by interacting nonlocal solitary waves, Physica D 101,270-288. Bound states ("multi-bump") solutions to both the FKdV and TNLS equations.
Calvo, D. C. and Akylas, T. R: 1997b, Stability of bound states near the zero-dispersion wavelength in optical fibers, Phys. Rev. E 56, 4757-4764. TNLS bions of minimum core-to-core separation are weakly unstable to infinitesimal perturbations but nonlinearity can stabilize weak perturbations so that long-time propagation as a bound state is possible in at least some parameter ranges.
Camassa, R: 1995, On the geometry of an atmospheric slow manifold, Physica D 84, 357-397. Nonlocal temporal oscillations.
Camassa, R and Tin, S.-K.: 1996, The global geometry of the slow manifold in the Lorenz-Krishnamurthy model, J. Atmos. Sci. 53, 3251-3264. Apply dynamical systems theory to prove existence of a locally slow manifold and nonexistence of a manifold that is globally slow.
Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A.: 1987, Spectml Methods for Fluid Dynamics, Springer-Verlag, New York.
Carnevale, G. F. and Vallis, G. K.: 1990, Pseudo-advective relaxation to stable states of inviscid two-dimensional fluids, J. Fluid Mech. 213, 549-57l. Numerical method for solving nonlinear eigenvalue problems for solitons.
534 WEAKLY NONLOCAL SOLITARY WAVES
Carr, J.: 1992, Slowly varying solutions of a nonlinear partial differential equation, in D. S. Broomhead and A. Iserles (eds), The Dynamics of Numerics and the Numerics of Dynamics, Oxford University Press, Oxford, pp. 23-30.
Carr, J. and Pego, R. L.: 1989, Metastable patterns in solutions of Ut = f. 2Uxx - f(u), Comm. Pure Appl. Math. 42, 523-576. Merger of fronts on an exponentially slow time scale.
Carrier, G. F. and Pearson, C. E.: 1968, Ordinary Differential Equations, Blaisdell, Waltham, MA. 229 pp.
Champneys, A. R.: 1997, Computation of homo clinic solutions to periodic orbits in a reduced water-wave problem, Physica D. Submitted. Review.
Champneys, A. R. and Groves, M. D.: 1997, A global investigation of solitarywave soutions to a two-parameter model for water waves, J. Fluid Mech. 342, 199-229. Numerical study of the bifurcations of a two-parameter FKdV equation with both classical and nonlocal solitons.
Champneys, A. R. and Lord, G. J.: 1997, Computation of homo clinic solutions to periodic orbits in a reduced water-wave problem, Physica D 102, 101-124. Submitted. "homo clinic solutions to periodic orbits" is a synonym for "weakly nonlocal solitary wave". Fourth-order, once-integrated version of the Fifth-Order Korteweg-deVries equation. Bound state ("multi-bump") solitons, both symmetric and asymmetric.
Chan, T. F.: 1984, Newton-like pseudo-arclength methods for computing simple turning points, SIAM J. Sci. Stat. Comput. 5, 135-148.
Chang, Y., Barcilon, A. and Blumsack, S.: 1994, An efficient method for investigating the flow evolution in shear layers, Geophys. Astrophys. Fluid Dyn. 76, 73-93. Apply the "modified dynamics" or "quenching method" of Vallis et al.(1989, 1990) and Shepherd(1990) to compute vortex structures in geophysical flows.
Chang, Y.-H.: 1991, Proof of an asymptotic symmetry of the rapidly forced pendulum, in H. Segur, S. Tanveer and H. Levine (eds), Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 213-22l.
Chang, Y.-H. and Segur, H.: 1991, An asymptotic symmetry of the rapidly forced pendulum, Physica D 51, 109-118. Beyond all orders perturbation theory in classical mechanics.
References 535
Charney, J. G. and Flierl, G. R: 1981, Oceanic analogues in large-scale atmospheric motions, in B. Warren and C. Wunsch (eds), Evolution of Physical Oceanography, MIT Press, Cambridge, pp. 502-546. Review.
Chen, G.-Y. and Boyd, J. P.: 1998, Propagation of nonlinear Kelvin wave packets in the equatorial ocean, J. Phys. Oceangr. Submitted. Superlinear dispersion of classical NLS packets.
Chester, W. and Breach, D. R: 1969, On the flow past a sphere at low Reynolds number, J. Fluid Mech. 37, 751-760. Log-and-power series.
Cheung, T. K. and Little, C. G.: 1990, Meteorological tower, microbarograph array, and solar observations of solitary waves in the nocturnal boundary layer, J. Atmos. Sci. 47, 2516-2536. Gravity waves, nonlocal through upward radiation from a leaky surface wave guide.
Christie, D. R: 1989, Long nonlinear waves in the lower atmosphere, J. Atmos. Sci. 46, 1462-1491. Numerical study of the Benjamin-Ono-Burgers equation with application to the Australian Morning Glory. Excellent review of the theory and observations of these nonlocal internal gravity waves.
Christie, D. R: 1992, The morning glory of the Gulf of Carpentaria: A paradigm for nonlinear waves in the lower atmosphere, Austral. Meteor. Mag. 41, 21-60. Comprehensive discussion of ducted internal gravity solitons, modeled by the Benjamin-Davis-Ono equation with radiative leaking to the upper atmosphere.
Christie, D. R, Muirhead, K. J. and Hales, A. L.: 1978, On solitary waves in the atmosphere, J. Atmos. Sci. 35, 805-825. Nonlocal internal gravity waves.
Christie, D. R, Muirhead, K. J. and Hales, A. L.: 1979, Intrusive density flows in the lower troposphere: A source of atmospheric solitons, J. Geophys. Res. 84, 4959-4970. Nonlocal gravity waves, which radiate from the surface duct to higher levels.
Christov, C. I. and Bekyarov, K. L.: 1990, A Fourier-series method for solving soliton problems, SIAM J. Sci. Stat. Comput. 11, 631-647. Rational functions on the infinite interval.
Chu, M. T.: 1988, On the continuous realization of iterative processes, SIAM Rev. 30, 375-387. Differential equations in pseudotime as models for Newton's and other iterations.
536 WEAKLY NONLOCAL SOLITARY WAVES
Cizek, J. and Vrscay, E. R.: 1982, Large order perturbation theory in the context of atomic and molecular physics - interdisciplinary aspects, Int. J. Quantum Chem. 21, 27-68.
Cizek, J., Damburg, R. J., Graffi, S., Grecchi, V., Il, E. M. H., Harris, J. G., Nakai, S., Paldus, J., Propin, R. K. and Silverstone, H. J.: 1986, 1/R expansion for Ht: Calculation of exponentially small terms and asymptotics, Phys. Rev. A 33, 12-54.
Clarke, R. A.: 1971, Solitary and cnoidal planetary waves, Geophys. Fluid Dyn. 2, 343-354. Rossby waves in a channel.
Clarke, R. H.: 1972, The morning glory: an atmospheric hydraulic jump, J. Appl. Meteor. 11, 304-311. Early paper on what is now thought be an undular bore, fissioning into nonlocal gravity solitons.
Clarke, R. H.: 1983a, Fair weather nocturnal inland wind surges and bores, Part 1. Nocturnal wind surges, Austral. Meteor. Mag. 31, 133-145.
Clarke, R. H.: 1983b, Fair weather nocturnal inland wind surges and bores, Part Il. Internal atmospheric bores in northern Australia, Austral. Meteor. Mag. 31, 147-160. Nonlocal internal gravity solitons.
Clarke, R. H.: 1984, Colliding sea breezes and atmospheric bores: twodimensional numerical studies, Austral. Meteor. Mag. 32, 207-226.
Clarke, R. H.: 1985, Geostrophic wind over Cape York Peninsula and pressure jumps around the Gulf of Carpentaria, Austral. Meteor. Mag. 33, 7-10.
Clarke, R. H.: 1986, Several atmospheric bores and a cold front over southern Australia, Austral. Meteor. Mag. 34, 65-76. Nonlocal internal gravity waves.
Clarke, R. H., Smith, R. K. and Reid, D. G.: 1981, The morning glory of the Gulf of Carpentaria: An atmospheric undular bore, Mon. Weather Rev. 109,1726-1750.
Cloot, A. and Weideman, J. A. C.: 1992, An adaptive algorithm for spectral computations on unbounded domains, J. Comput. Phys. 102, 398-406.
Cloot, A., Herbst, B. M. and Weideman, J. A. C.: 1990, A numerical study of the cubic-quintic Schrodinger equation, J. Comput. Phys. 86, 127-146.
Combescot, R., Dombe, T., Hakim, V. and Pomeau, Y.: 1986, Shape selection of SafIman-Taylor fingers, Phys. Rev. Letters 56, 2036-2039.
References 537
Conte, S. D. and de Boor, C.: 1980, Elementary Numerical Analysis, 3 edn, McGraw-Hill, New York. 250 pp.
Costin, 0.: 1995, Exponential asymptotics, trans-series and generalized Borel summation for analytic nonlinear rank one systems of ODE's, International Mathematics Research Notices 8, 377-418.
Costin, 0.: 1998, On Borel summation and stokes phenomenon for nonlinear rank one systems of ODE's, Duke Math. J. To appear. Connections with Berry smoothing and Ecalle resurgence.
Costin, o. and Kruskal, M. D.: 1996, Optimal uniform estimates and rigorous asymptotics beyond all orders for a class of ordinary differential equations, Proc. Roy. Soc. London A 452, 1057-1085.
Costin, o. and Kruskal, M. D.: 1998, On optimal truncation of divergent series solutions of nonlinear differential systems; berry smoothing, Proc. Roy. Soc. London A. Submitted. Rigorous proofs of some assertions and conclusions of Berry's smoothing of discontinuities in Stokes phenomenon.
Courtier, P. and Talagrand, 0.: 1990, Variational assimilation of meteorological observations with the direct and adjoint shallow-water equations., Tellus A 42, 531-549. Variational/optimal control methods for initialization in numerical weather prediction.
Coustias, E. A. and Segur, H.: 1991, A new formulation for dendritic crystal growth in two dimensions, in H. Segur, S. Tanveer and H. Levine (eds) , Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 87-104.
Cox, S. M.: 1991, Two-dimensional flow of a viscous fluid in a channel with porous walls, J. Fluid Mech. 227, 1-33. Multiple solutions differing by exponentially small terms.
Cox, S. M. and King, A. C.: 1997, On the asymptotic solution of a highorder nonlinear ordinary differential equation, Proc. Roy. Soc. London A 453, 711-728. Berman-Terrill-Robinson problem with good review of earlier work.
Crook, N. A.: 1984, The formation of the Morning Glory, in D. K. Lilly and T. Gal-Chen (eds), Mesoscale Meteorology - Theories, Observations and Models, Reidel, Dordrecht, pp. 349-353. Nonlocal horizontally-ducted gravity waves.
Crook, N. A.: 1986, The effect of ambient stratification and moisture on the motion of atmospheric undular bores, J. Atmos. Sci. 43, 171-181. Nonlocal
538 WEAKLY NONLOCAL SOLITARY WAVES
gravity waves, such as the Australian Morning Glory and thunderstormgenerated solitons.
Crook, N. A.: 1988, Trapping of low-level internal gravity waves, J. Atmos. Sci. 45(10), 1533-1541. Vertical trapping of horizontally-ducted nonlocal gravity wave solitons.
Crook, N. A. and Miller, M. J.: 1985, A numerical and analytical study of atmospheric undular bores, Quart. J. Royal Meteor. Soc. 111, 225-242. Nonlocal gravity waves, such as the Australian Morning Glory and thunderstorm-generated solitons.
Dai, S., Sigalov, G. F. and Diogenov, A. V.: 1990, Science in China A 33, 843-853. Analytic approximate solutions to the FKdV equation.
Daley, R: 1991, Atmospheric Data Analysis, Cambridge University Press, New York. No solitons, but good background on the "slow manifold".
Dashen, R F., Hasslacher, B. and Neveu, A.: 1975, Particle spectrum in model field theories from semiclassical functional integral techniques, Phys. Rev. D 11, 3424-3450. 1;4 breathers; multiple scales perturbation theory.
Davidenko, D.: 1953a, On a new method of numerically integrating a system of nonlinear equations, Doklady Akademie Nauk SSSR 88, 601-604.
Davidenko, D.: 1953b, On the approximate solution of a system of nonlinear equations, Ukraine Mat. Zurnal5, 196-206.
Davfs, P. J.: 1975, Interpolation and Approximation, Dover Publications, New York. 200 pp.
Davis, R E. and Acrivos, A.: 1967, Solitary internal waves in deep water, J. Fluid Mech. 29, 593-607.
de Bruijn, N. G.: 1981, Asymptotic Methods in Analysis, 3 edn, Dover, New York. Mixed series of logarithms and powers, illustrated by a cousin of the Lambert W-function. Nothing about nonlocal solitons.
Decker, D. W. and Keller, H. B.: 1980, Path following near bifurcation, Comm. Pure Appl. Math. 34, 149-175. Solving systems of nonlinear equations and shooting the bifurcation point.
Dennis, Jr., J. E. and Schnabel, R B.: 1983, Numerical Methods for Nonlinear Equations and Unconstrained Optimization, Prentice-Hall, Englewood Cliffs, New Jersey.
References 539
Dias, F.: 1994, Capillary-gravity periodic and solitary waves, Phys. Fluids 6, 2239-2241.
Dias, F., Menasce, D. and Vanden-Broeck, J.-M.: 1996, Numerical study of capillary-gravity solitary waves, Europ. J. Mech. B 15, 17-36.
Dingle, R. B.: 1956, The method of comparison equations in the solution of linear second-order differential equations(generalized W. K. B. method), Applied Scientific Research B 5, 345-367.
Dingle, R. B.: 1958a, Asymptotic expansions and converging factors I. General theory and basic converging factors, Proc. Roy. Soc. London A 244, 456-475.
Dingle, R. B.: 1958b, Asymptotic expansions and converging factors IV. Confluent hypergeometric, parabolic cylinder, modified Bessel and ordinary Bessel functions, Proc. Roy. Soc. London A 249, 270-283.
Dingle, R. B.: 1958c, Asymptotic expansions and converging factors Il. Error, Dawson, Fresnel, exponential, sine and cosine, and similar integrals, Proc. Roy. Soc. London A 244, 476-483.
Dingle, R. B.: 1958d, Asymptotic expansions and converging factors V. Lommel, Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions, Proc. Roy. Soc. London A 249, 284-292.
Dingle, R. B.: 1958e, Asymptotic expansions and converging factors Ill. Gamma, psi and polygamma functions, and Fermi-Dirac and BoseEinstein integrals, Proc. Roy. Soc. London A 244, 484-490.
Dingle, R. B.: 1958f, Asymptotic expansions and converging factors VI. Application to physical prediction, Proc. Roy. Soc. London A 249, 293-295.
Dingle, R. B.: 1973, Asymptotic Expansions: Their Derivation and Interpretation, Academic, New York. Beyond all orders asymptotics.
Dodd, R. K., Eilbeck, J. C., Gibbon, J. D. and Morris, H. C.: 1982, Solitons and Nonlinear Wave Equations, Academic Press, New York. Good, comprehensive, practical review of classical solitons. 600 pp.
Doedel, E., Keller, H. B. and Kernevez, J. P.: 1991a, Numerical analysis and control of bifurcation problems (I) Bifurcation in finite dimensions, Internat. J. Bifurcation Chaos 1, 493-520. Good review of pseudoarclength continuation and the tracking of bifurcating branches to nonlinear equations.
540 WEAKLY NONLOCAL SOLITARY WAVES
Doedel, E., Keller, H. B. and Kernevez, J. P.: 1991b, Numerical analysis and control of bifurcation problems (II) Bifurcation in infinite dimensions, Internat. J. Bifurcation Chaos 1, 745-772.
Donaldson, J. D. and Elliott, D.: 1972, Estimating contour integrals, SIAM J. Numer. Anal. 9, 573-602. Exponential smallness in numerical analysis (implicitly) .
Doviak, R J. and Christie, D. R: 1989, Thunderstorm generated solitary waves: A wind shear hazard, J. Aircraft 26, 423-431. Nonlocal gravity solitons.
Doviak, R J. and Ge, R: 1984, An atmospheric solitary gust observed with a Doppler radar, a tall tower and a surface network, J. Atmos. Sci. 41, 2559-2573.
Doviak, R J., Chen, S. S. and Christie, D. R: 1991, A thunderstorm-generated solitary wave observation compared with theory for nonlinear waves in a sheared atmosphere, J. Atmos. Sci. 48(1), 87-111.
Doviak, R. J., Thomas, K. W. and Christie, D. R: 1989, The wavefront shape, position and evolution of a great solitary wave of translation., IEEE Trans. Geoscience and Remote Sensing 27(6), 658-665.
Drake, V. A.: 1984, A solitary wave disturbance of the marine boundary layer over Spencer Gulf revealed by radar observations of migrating insects, Austral. Meteor. Mag. 32, 131-135.
Drake, V. A.: 1985, Solitary wave disturbances of the nocturnal boundary layer revealed by radar observations of migrating insects, Boundary-layer Meteorology 31, 269-286.
Dumas, H. S.: 1991, Existence and stability of particle channeling in crystals on timescales beyond all orders, in H. Segur, S. Tanveer and H. Levine (eds), Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 267-273.
Dumas, H. S.: 1993, A Nekhoroshev-like theory of classical particle channeling in perfect crystals, Dynamics Reported 2, 69-115. Beyond all orders perturbation theory in crystal physics.
Dumas, H. S. and Ellison, J. A.: 1991, Nekhoroshev's theorem, ergodicity, and the motion of energetic charged particles in crystals, in J. A. Ellison and H. Uberall (eds), Essays on Classical and Quantum Dynamics, Gordon and Breach, Philadelphia, pp. 17-56.
References 541
Dunster, T. M.: 1996, Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one, Methods and Applications of Analysis 3(1),109-134.
Durran, D. R.: 1991, The third-order Adams-Bashforth method: an attractive alternative to leapfrog time-differencing, Mon. Weather Rev. 119, 702-720. Review of time-integration algorithms; good discussion of both dispersion and dissipation errors for propagating waves.
D'yakonov, E. G.: 1961, An iteration scheme, Dokl. Akad. Nauk SSSR 138, 522-. Numerical algorithm.
Dym, H. and McKean, H. P.: 1972, Fourier Series and Integrals, Academic Press, New York. 129 pp.
Dyson, F. J.: 1952, Divergence of perturbation theory in quantum electrodynamics, Phys. Rev. 85, 631-632.
Ecalle, J.: 1981, Les fonctions nffsurgentes, Universite de Paris-Sud, Paris. Beyond all orders perturbation theory.
Eckhaus, w.: 1992a, On water waves of Froude number slightly higher than one and Bond number less than 1/3, ZAMP 43, 254-269. Proof of exponential smallness of the oscillatory wings for FKdV nanopterons.
Eckhaus, W.: 1992b, Singular perturbations of homo clinic orbits in R4, SIAM J. Math. Anal. 23, 1269-1290.
Egger, J.: 1984, On the theory of the morning glory, Beitrage Physik Atmosphere 57,123-134.
Eilbeck, J. C. and Flesch, R.: 1990, Calculation of families of solitary waves on discrete lattices, Phys. Lett. A 149, 200--202. Fourier cosine collocation to compute solitary waves which propagate as continuous functions of time on a discrete lattice of interacting masses.
Elliott, D.: 1964, The evaluation and estimation of the coefficients in the Chebyshev series expansion of a function, Math. Comp. 18, 274-284. This and the next two papers are classic contributions to the asymptotic theory of Chebyshev coefficients.
Elliott, D.: 1965, Truncation errors in two Chebyshev series approximations, Math. Comp. 19, 234-248. Errors in Lagrangian interpolation with a general contour integral representation and an exact analytical formula for l/(a + x).
542 WEAKLY NONLOCAL SOLITARY WAVES
Elliott, D. and Szekeres, G.: 1965, Some estimates of the coefficients in the Chebyshev expansion of a function, Math. Comp. 19, 25-32. The Chebyshev coefficients are exponentially small in the degree n.
Eydeland, A. and Thrkington, B.: 1987, A computational study of Rossby solitary waves, Stud. Appl. Math. 76, 37-67. Non-Newtonian numerical method for computing classical solitary waves of quasi-geostrophic flow.
Eydeland, A. and Thrkington, B.: 1988, A computational method of solving free-boundary problems in vortex dynamics, J. Comput. Phys. 78, 194-214. Non-Newtonian numerical method for computing classical solitary waves.
Falques, A. and Iranzo, V.: 1992, Edge waves on a longshore shear flow, Phys. Fluids pp. 2169-2190. Rational Chebyshev and Laguerre pseudospectral methods on a semi-infinite domain.
Farmer, D. M. and Smith, J. D.: 1980, Tidal interaction of stratified flow with a sill in the Knight inlet, Deep Sea Res. A 27, 239-254. Observations of mode-2 internal gravity solitons followed by small amplitude mode-2 wavetrain as predicted by Akylas and Grimshaw(1992).
Fedorov, K. N. and Ginsburg, A. I.: 1986, Mushroom-like currents (vortex dipoles) in the ocean and a laboratory tank., Annales Geophys. 4, 507-516. Observations of modon-like vortex pairs.
Finlayson, B. A.: 1973, The Method of Weighted Residuals and Variational Principles, Academic, New York. Pseudospectral method; 412 pp.
Finlayson, B. A. and Scriven, L. E.: 1966, The method of mean weighted residuals - a review, Appl. Mech. Revs. 12, 735-748. Pseudospectral method.
Flierl, G. R.: 1979, Baroclinic solitary waves with radial symmetry, Dyn. Atmos. Oceans 3, 15-38. Flierl-Petviashvili monopoles (classical, radially symmetric soliton).
Flierl, G. R.: 1987, Isolated eddy models in geophysics, Ann. Rev. Fluid Mech. 19, 493-530.
Flierl, G. R.: 1994, Rings: Semicoherent oceanic features, Chaos 4, 355-367. Review; radiative leakage in ocean vortices.
Flierl, G. R., Larichev, V. D., McWilliams, J. C. and Reznik, G. M.: 1980, The dynamics of baroclinic and barotropic solitary eddies, Dyn. Atmos. Oceans 5, 1-41. Classical modons.
References 543
Flierl, G. R, Stern, M. E. and Whitehead, J. A.: 1983, The physical significance of modons: laboratory experiments and general integral constraints, Dyn. Atmospheres Oceans 7, 233-263. Prove monopoles cannot be classical solitons; good rotating tank experiments.
FIar, J. B. and van Heijst, G. J. F.: 1996, Stable and unstable monopolar vortices in a stratified fluid, J. Fluid Mech. 311, 257-287.
Fornberg, B.: 1996, A Practical Guide to Pseudospectral Methods, Cambridge University Press, New York.
Fornberg, B. and Sloan, D.: 1994, A review of pseudospectral methods for solving partial differential equations, in A. Iserles (ed.), Acta Numerica, Cambridge University Press, New York, pp. 203-267.
Fornberg, B. and Whitham, G. B.: 1978, A numerical and theoretical study of certain nonlinear wave phenomena, Phil. Trans. Royal Soc. London 289,373-404. Efficient Fourier pseudospectral method for wave equations; the linear terms, which have constant coefficients, are integrated in time exactly.
Fowler, A. C. and Kember, G.: 1996, On the Lorenz-Krishnamurthy slow manifold, J. Atmos. Sci. 53, 1433-1437. Slow manifold of the forced-anddamped LK Quintet.
Froman, N.: 1966, The energy levels of double-well potentials, Arkiv for Fysik 32{ 4), 79-96. WKB method for exponentially small splitting of eigenvalue degeneracy.
Fulton, R, Zrnic, D. S. and Doviak, R J.: 1990, Initiation of a solitary wave family in the demise of a nocturnal thunderstorm density current, J. Atmos. Sci. 47, 319-337. Nonlocal through upward radiation.
Funaro, D.: 1992, Polynomial Approximation of Differential Equations, Springer-Verlag, New York. 313 pp.; monograph on Chebyshev and other spectral methods. Numerical.
Funaro, D. and Kavian, 0.: 1991, Approximation of some diffusion evolution equations in unbounded domains by Hermite functions, Math. Comp. 57, 597-619. Numerical.
Fusco, G. and Hale, J. K.: 1989, Slow motion manifolds, dormant instability and singular perturbations, Journal of Dynamics and Differential Equations 1, 75-94. Exponentially slow frontal motion.
544 WEAKLY NONLOCAL SOLITARY WAVES
Galkin, V. M. and Stepanyants, Y. A.: 1991, On the existence of stationary solitary waves in a rotating fluid, J. Appl. Math. Meeh. 55, 939-943. Proves nonexistence of classical solitary waves for the RMKdV equation, here called the "Ostrovsky equation" .
Garda-Archilla, B.: 1996, A spectral method for the equal width equation, J. Comput. Phys. 125, 395-402. Time-integration with Fourier pseudospectral spatial discretization for a wave equation.
Geicke, J.: 1994, Logarithmic decay of cjJ4 breathers of energy e ::; 1, Phys. Rev. E 49, 3539-3542.
Gilman, O. A., Grimshaw, R. and Stepanyants, Y. A.: 1995, Approximate analytical and numerical solutions of the stationary Ostrovsky equation, Stud. Appl. Math. 95, 115-126. "Ostrovskyequation" is a synonym for the Rotation-Modified Korteweg-deVries (RMKdV) equation. Their periodic solutions are the imbrication of parabolic arcs, smoothed at the crests by an inner approximation which is the imbrication of Korteweg-de Vries solitons.
Gilman, O. A., Grimshaw, R. and Stepanyants, Y. A.: 1996, Dynamics of internal solitary waves in a rotating fluid, Dyn. Atmos. Oceans 23, 403-411. Further study of the RMKdV (Ostrovsky) equation with some initial value calculations.
Gingold, H. and Hu, J.: 1991, Transcendentally small reflection of waves for problems with/without turning points near infinity: A new uniform approach, Journal of Mathematical Physics 32(12), 3278-3284. Generalized WBK (Liouville-Green) for above-the-barrier scattering.
Glowinski, R., Keller, H. B. and Reinhart, L.: 1985, Continuation conjugate gradient methods for the least squares solution of nonlinear boundary value problems, SIAM J. Sci. Stat. Comput. 6, 793-832. Not solitons, but interesting numerical methods for solving nonlinear boundary value problems, such as those for solitons.
Gollub, J. P.: 1991, An experimental assessment of continuum models of dendritic growth, in H. Segur, S. Tanveer and H. Levine (eds) , Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 76-86.
Goncharov, V. P. and Matveyev, A. K.: 1982, Observations of non-linear waves on an atmospheric inversion, USSR Atmospheric and Oceanic Physics 18, 61-64. Nonlocal gravity solitons.
References 545
Gorshkov, K. A. and Ostrovsky, L. A.: 1981, Interactions of solitons in noninterable systems: direct perturbation method and applications, Physica D 3, 428-438. No discussion of nonlocal solitary waves, but a good review of methods used by Grimshaw and Malomed t(1993) to develop the first theory for bound states of weakly nonlocal solitary waves.
Gorshkov, K. A. and Papko, v. v.: 1977, Dynamic and stochastic oscillations of soliton lattices, Soviet Physics JETP 46, 92-97.
Gorshkov, K. A., Ostrovskii, L. A. and Papko, V. v.: 1976, Interactions and bound states of solitons as classical particles, Soviet Physics JETP 44, 306-311. Nothing about nonlocal solitary waves, but good discussion of perturbation theory for interacting solitary waves.
Gorshkov, K. A., Ostrovskii, L. A. and Papko, V. v.: 1977, Soliton turbulence in a system with weak dispersion, Soviet Physics Doklady 22, 378-380.
Gorshkov, K. A., Ostrovskii, L. A., Papko, V. V. and Pikovsky, A. S.: 1979, On the existence of stationary multisolitons, Phys. Lett. A 74, 177-179.
Gottlieb, D. and Orszag, S. A.: 1977, Numerical Analysis of Spectral Methods, SIAM, Philadelphia, PA. 200 pp.
Gradshteyn, I. S. and Ryzhik, I. M.: 1965, Table of Integrals, Series, and Products, 4 edn, Academic Press, New York. 1086 pp.
Grasman, J. and Matkowsky, B. J.: 1976, A variational approach to singularly perturbed boundary value problems for ordinary and partial differential equations with turning points, SIAM J. Appl. Math. 32, 588-597. Resolve the failure of standard matched asymptotics for the problem of Ackerberg and O'Malley (1970) by applying a non-perturbative variational principle; MacGillivray (1997) solves the same problem by incorporating exponentially small terms into matched asymptotics.
Greatbatch, R. J.: 1985, Kelvin wave fronts, Rossby solitary waves and nonlinear spinup of the equatorial oceans, J. Geophys. Res. 90, 9097-9107. Shows how easy it is to generate large-scale Rossby waves from very un-solitonic initial conditions and wind stresses.
Grimshaw, R.: 1986, Theory of solitary waves in shallow fluids, in N. P. Cheremisinoff (ed.), Encyclopedia of Fluids Mechanics Volume 2, Gulf Publishing Company, Houston, pp. 3-25.
Grimshaw, R.: 1994a, Exponential asymptotics in the reduced KuramotoSivashinsky equation., in P. L. Sachdev and R. E. Grundy (eds) , Nonlinear Diffusion Phenomena: Proceedings of Meeting at Bangalore, India, 1992, Narosa Publishing, pp. 51-67.
546 WEAKLY NONLOCAL SOLITARY WAVES
Grimshaw, R.: 1994b, Generation of solitary waves by external forcing, Geometrical Methods in Fluid Dynamics, number 94-12 in WHOI Reports, Woods Hole Oceanographic Institute, Woods Hole Oceanographic Institute, Woods Hole, Massachusetts, pp. 283-291. Review.
Grimshaw, R.: 1994c, Solitary waves with oscillatory tails and exponential asymptotics, Geometrical Methods in Fluid Dynamics, number 94-12 in WHOI Reports, Woods Hole Oceanographic Institute, Woods Hole Oceanographic Institute, Woods Hole, Massachusetts, pp. 292-298. Review.
Grimshaw, R.: 1997, Internal solitary waves, in P. L. Liu (ed.), Advances in Coastal and Ocean Engineering, World Scientific Publishing, Singapore, pp. 1-30. Review; good discussion of several species of nonlocal solitary waves.
Grimshaw, R. and Malomed, B. A.: 1995, Nonexistence of gap solitons in nonlinearly coupled systems, Phys. Lett. A 198,205-208. Coupled pair of KdV equations; solitary waves propagating in opposite directions become nonlocal solitary waves when the coupling is turned on.
Grimshaw, R. and Tang, S.: 1990, The Rotation-Modified KadomtsevPetviashvili equation: an analytical and numerical study, Stud. Appl. Math. 83, 223-248. Kelvin-like solitons of the RMKP equation radiate Poincare waves.
Grimshaw, R. and Zhu, Y.: 1994, Oblique interactions between internal solitary waves, Stud. Appl. Math. 92, 249-270. Theory; their analysis is applied to colliding Morning Glories in Reeder et al.(1995).
Grimshaw, R. H. J.: 1981, Evolution equations for long, nonlinear internal waves in stratified shear flows, Stud. Appl. Math. 65, 159-188. Radiative damping, friction, and temporal and spatial variations when gravity waves are ducted, as in the Australian Morning Glory, in a waveguide of very stable air at the surface surmounted by very weakly stratified fluid at higher levels.
Grimshaw, R. H. J.: 1985, Evolution equations for weakly nonlinear, long internal waves in a rotating fluid, Stud. Appl. Math. 73, 1-33. Derivation of the Rotation-Modified Korteweg-deVries (shallow water) and RotationModified Benjamin-Ono (deep water) equations; analysis of both strong rotation and weak rotation limits.
Grimshaw, R. H. J.: 1992, The use of Borel-summation in the establishment of non-existence of certain travelling-wave solutions of the KuramotoSivashinskyequation, Wave Motion 15,393-395.
References 547
Grimshaw, R. H. J.: 1995, Weakly nonlocal solitary waves in a singularly perturbed nonlinear Schroedinger equation, Stud. Appl. Math. 94, 257-270. Third-Order Nonlinear Schroedinger (TNLS) equation through complex plane-matched asymptotics and Borel summation.
Grimshaw, R. H. J., Afanasjev, V. V. and Kivshar, Y. S.: 1997a, Dark solitons with nonvanishing oscillating tails, Phys. Lett. A. Submitted.
Grimshaw, R. H. J. and Joshi, N.: 1995, Weakly non-local solitary waves in a singularly-perturbed Korteweg-deVries equation, SIAM J. Appl. Math. 55, 124-135.
Grimshaw, R. H. J. and Malomed, B. A.: 1993, A note on the interaction between solitary waves in a singularly-perturbed Korteweg-deVries equation, J. Phys. A 26, 4087-4091. Predicted the existence of bound states of solitons (bions, trions, etc.) for the FKdV equation as later confirmed numerically by Boyd(1996e).
Grimshaw, R. H. J. and Melville, W. K.: 1989, On the derivation of the modified Kadomtsev-Petviashvili equation, Stud. Appl. Math. 80, 183-202. Show that the constraint on initial conditions for RMKP equation is unphysical in the sense that the solitons are weakly nonlocal and radiate energy far from the core through Poincare waves.
Grimshaw, R. H. J., He, J.-M. and Ostrovsky, L. A.: 1997b, Terminal damping of a solitary wave due to radiation in rotational systems., Stud. Appl. Math. Accepted. RMKdV solitons collapse into linear, dispersing wave packets in finite time, as shown by analysis, but a soliton then reforms from the leading edge of the packet and this death-and-birth cycle recurrs repeatedly, as shown by numerical solutions.
Grimshaw, R., Malomed, B. A. and Tian, X.: 1995, Gap-soliton hunt in a coupled Korteweg-deVries system, Phys. Lett. A 201, 285-292.
Grimshaw, R., Ostrovsky, L. A., Shrira, V. I. and Stepanyants, Y. A.: 1997c, Nonlinear surface and internal gravity waves in a rotating ocean, Surveys in Geophysics. Accepted. RMKdV and RMKP equations. Review.
Grosch, C. E. and Orszag, S. A.: 1977, Numerical solution of problems in unbounded regions: coordinate transforms, J. Comput. Phys. 25, 273-296. Numerical.
Grundy, R. E. and Allen, H. R.: 1994, The asymptotic solution of a family of boundary value problems involving exponentially small terms, IMA J. Appl. Math. 53, 151-168.
548 WEAKLY NONLOCAL SOLITARY WAVES
Guillou, J. C. L. and Zinn-Justin, J. (eds): 1990, Large-Order Behaviour of Perturbation Theory, North-Holland, Amsterdam. Exponential corrections to power series, mostly in quantum mechanics.
Haase, S. P. and Smith, R. K.: 1984, Morning glory wave clouds in Oklahoma: A case study, Mon. Weather Rev. 112, 2078-2089. Nonlocal gravity solitons.
Haines, K.: 1989, Baroclinic modons as prototypes for atmospheric blocking, J. Atmos. Sci. 46, 3202-3218. Classical modons in shear flow.
Hakim, V.: 1991, Computation of transcendental effects in growth problems: Linear solvability conditions and nonlinear methods - the example of the geometric model, in H. Segur, S. Tanveer and H. Levine (eds), Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 15-28.
Hakim, V. and Mallick, K.: 1993, Exponentially small splitting of separatrices, matching in the complex plane and Borel summation, Nonlinearity 6,57-70. Very readable analysis.
Hale, J. K.: 1992, Dynamics and numerics, in D. S. Broomhead and A. Iserles (eds) , The Dynamics of Numerics and the Numerics of Dynamics, Oxford University Press, Oxford, pp. 243-254.
Hammersley, J. M. and Mazzarino, G.: 1991, Numerical analysis of the geometric model for dendritic growth of crystals, in H. Segur, S. Tanveer and H. Levine (eds), Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 37-66.
Hanson, F. B.: 1990, Singular point and exponential analysis, in R. Wong (ed.), Asymptotic and Computational Analysis, Marcel Dekker, New York, pp. 211-240.
Hardy, G. H.: 1949, Divergent Series, Oxford University Press, New York.
Harrell, E. M.: 1978, On the asymptotic rate of eigenvalue degeneracy, Commun. Math. Phys. 60, 73-95.
Harrell, E. M.: 1980, Double wells, Commun. Math. Phys. 75, 239-261. Exponentially small splitting of eigenvalues.
Hasegawa, A.: 1989, Optical Solitons in Fibers, Springer-Verlag, New York.
Hasse, S. P. and Smith, R. K.: 1989, The numerical simulation of atmospheric gravity currents, Part H. Environments with stable layers, Geophys. Astrophys. Fluid Dyn. 46, 35-51. Generation mechanisms for weakly nonlocal solitary waves.
References 549
Haupt, R. L. and Haupt, S. E.: 1997, Practical Genetic Algorithms, John Wiley, New York. One example is the computation of cnoidal and double cnoidal waves of the FKdV equation by combining the Fourier pseudospectral method with a genetic algorithm.
Haupt, S. E. and Boyd, J. P.: 1988, Modeling nonlinear resonance: A modification to Stokes' perturbation expansion, Wave Motion 10,83-98. These resonances, analyzed here for small amplitude, become nonlocal solitons at large amplitude. The coefficient of cos(3X) in the fifth order solution should include a term proportional to -8970113 , not -9870113 as printed. The assertion that there are only two roots for wavenumber three resonance is incorrect; there are actually three roots (Chapter 5).
Haupt, S. E. and Boyd, J. P.: 1991, Double cnoidal waves of the KortewegdeVries equation: The boundary value approach, Physica D 50,117-134.
Haupt, S. E., McWilliams, J. C. and Tribbia, J. J.: 1993, Modons in shear flow, J. Atmos. Sci. 50(9), 1181-1198. Classical modons.
Heading, J.: 1962, An Introduction to Phase Integral Methods, Wiley, New York. Very readable, short primer on classical WKB theory.
Henderson, M. E. and Keller, H. B.: 1990, Complex bifurcation from real paths, SIAM J. Appl. Math. 50(2),460-482. Solving systems of nonlinear algebraic equations.
Herbst, B. M. and Ablowitz, M.: 1992, Numerical homo clinic instabilities in the sine-Gordon equation, Quaestiones Mathematicae 15, 345-363.
Herbst, B. M. and Ablowitz, M.: 1993, Numerical chaos, symplectic integrators, and exponentially small splitting distances, J. Comput. Phys. 105, 122-132.
Hildebrand, F. H.: 1974, Introduction to Numerical Analysis, Dover, New York. Numerical; asymptotic-but-divergent series in h for errors.
Hinton, D. B. and Shaw, J. K.: 1985, Absolutely continuous spectra of 2d order differential operators with short and long range potentials, Quart. J. Math. 36, 183-. Exponential smallness in eigenvalues.
Hobbs, A. K., Kath, W. L. and Kreigsmann, G. A.: 1991, Bending losses in optical fibers, in H. Segur, S. Tanveer and H. Levine (eds) , Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 309-316.
Holland, J. H.: 1975, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor. Ur-text on genetic algorithms.
550 WEAKLY NONLOCAL SOLITARY WAVES
Holmes, P., Marsden, J. and Scheurle, J.: 1988, Exponentially small splitting of separatrices with applications to KAM theory and degenerate bifurcations, Comtempomry Mathematics 81,214-244.
Hong, D. C. and Langer, J. S.: 1986, Analytic theory of the selection mechanism in the Saffman-Taylor problem, Phys. Rev. Letters 56, 2032-2035.
Hopfinger, E. J. and van Heijst, G. J. F.: 1991, Vortices in rotating fluids, Ann. Rev. Fluid Mech. 25, 241-289. Rossby solitons and modons.
Howls, C. J.: 1992, Hyperasymptotics for integrals for finite endpoints, Proc. Roy. Soc. London A 439,373-396.
Howls, C. J.: 1997, Hyperasymptotics for multidimensional integrals, exact remainder terms and the global connection problem, Proc. Roy. Soc. London A 453, 2271-2294.
Hu, J.: 1996, Asymptotics beyond all orders for a certain type of nonlinear oscillator, Stud. Appl. Math. 96,85-109.
Hu, J. and Kruskal, M. D.: 1991, Reflection coefficient beyond all orders for singular problems, in H. Segur, S. Tanveer and H. Levine (eds), Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 247-253.
Hukuda, H.: 1979, Solitary Rossby waves in a two-layer system, Tellus 31,161-169.
Hunkins, K and Fliegel, M.: 1973, Internal undular surges in Seneca Lake: a natural occurence of solitons, J. Geophys. Res. 78, 539-548. These solitons radiatively decay through the barotropic and other baroclinic modes.
Hunter, J. K: 1990, Numerical solutions of some nonlinear dispersive wave equations, Lectures in Applied Mathematics 26, 301-316. Derives, using matched asymptotics, an approximation to a micropteron in one space dimension; RMKdV equation.
Hunter, J. K and Scheurle, J.: 1988, Existence of perturbed solitary wave solutions to a model equation for water waves, Physica D 32, 253-268. FKdV nonlocal solitons.
Hunter, J. K and Vanden-Broeck, J.-M.: 1983, Solitary and periodic gravitycapillary waves of finite amplitude, J. Fluid Mech. 134, 205-219.
If, F., Berg, P., Christiansen, P. L. and Skovgaard, 0.: 1987, Split-step spectral method for nonlinear Schroedinger equation with absorbing boundaries, J. Comput. Phys. 72,501-503. An artificial damping "Y(x) is added to make
References 551
the NLS equation iUt + !uxx + lul 2u = -iJ(x)u; 'Y(x) = 'Yo{sech2(a[xL/2]) + sech2(a[x + L/2])} to create absorbing boundaries.
1ngersoll, A. P.: 1973, Jupiter's Great Red Spot: A free atmospheric vortex?, Science 182, 1346-1348. Rossby soliton.
1ngersoll, A. P. and Cuong, P. G.: 1981, Numerical model of long-lived Jovian vortices, J. Atmos. Sci. 38, 2067-2076. Numerical calculations of Rossby solitons in shear flow.
1ooss, G. and Kirchgiissner, K.: 1992, Water-waves for small surface tension -an approach via normal forms, Proc. Royal Soc. Edinburgh A 122,267-299. Prove the existence of FKdV nanopterons using center manifold theory.
1skandarani, M., Haidvogel, D. and Boyd, J. P.: 1995, A staggered spectral finite element method for the shallow water equations, Int. J. Num. Meths. Fluids 20, 393-414. Test case includes equatorial Rossby solitons.
Jardine, M., AlIen, H. R, Grundy, R E. and Priest, E. R: 1992, A family of two-dimensional nonlinear solutions for magnetic field annihilation, J. Geophys. Res. - Space Physics 97, 4199-4207.
Jones, D. S.: 1966, Fourier transforms and the method of stationary phase, Journal of the Institute for Mathematics and its Applications 2, 197-222.
Jones, D. S.: 1990, Uniform asymptotic remainders, in R Wong (ed.), Asymptotic Comput. Anal., Marcel Dekker, New York, pp. 241-264.
Jones, D. S.: 1993, Asymptotic series and remainders, in B. D. Sleeman and R J. Jarvis (eds) , Ordinary and Partial Differential Equations, Volume IV, Longman, London, pp. 12-.
Jones, D. S.: 1994, SIAM J. Math. Anal. 25, 474-490. Hyperasymptotics.
Jones, D. S.: 1997, Introduction to asymp to tics: a treatment using nonstandard analysis, World Scientific, Singapore. 160 pp.; includes a chapter on hyperasymptotics.
Kakutani, T. and Yamasaki, N.: 1978, Solitary waves on a two-layer fluid., J. Phys. Soc. Japan 45,674-679. KdV equation with mixed cubic/quadratic nonlinearity; does not discuss nonlocal solitons.
Kaplun, S.: 1957, Low Reynolds number flow past a circular cylinder, J. Math. Mech. 6, 595-603. Log-plus-power expansions.
552 WEAKLY NONLOCAL SOLITARY WAVES
Kaplun, S.: 1967, Fluid Mechanics and Singular Perturbations, Academic Press, New York. ed. by P. A. Lagerstrom, L. N. Howard and C. S. Liu; Analyzed difficulties of log-plus-power expansions.
Kaplun, S. and Lagerstrom, P. A.: 1957, Asymptotic expansions of NavierStokes solutions for small Reynolds number, J. Math. Mech. 6, 585-593.
Karpman, V. 1.: 1993, Radiation by solitons due to higher-order dispersion, Phys. Rev. E 47, 2073-2082.
Karpman, V. 1.: 1994a, Stationary solitary waves of the fifth-order KdV-type equation, Phys. Lett. A 186, 300-302.
Karpman, V. 1.: 1994b, Radiating solitons of the fifth-order KdV-type equation, Phys. Lett. A 186, 303-308.
Karpman, V. 1. and Solov'ev, V. v.: 1981, A perturbation theory for soliton systems, Physica D 1, 142-164. No weakly nonlocal solitons per se.
Kasahara, A.: 1977, Numerical integration of the global barotropic primitive equations with Hough harmonic expansions, J. Atmos. Sci. 34, 687-701. Hough spectral basis, which provides the mathematical framework for the slow manifold.
Kasahara, A.: 1978, Further studies on a spectral model of the global barotropic primitive equations with Hough harmonic expansions, J. Atmos. Sci. 35, 2043-2051.
Kath, W. L. and Kriegsmann, G. A.: 1988, Optical tunnelling: radiation losses in bent fibre-optic waveguides, [MA J. Appl. Math. 41, 85-103. Radiation loss is exponentially small in the small parameter, so "beyond all orders" perturbation theory is developed here.
Katopodes, N. D., Sanders, B. F. and Boyd, J. P.: 1998, Short wave behavior of long wave equations, Journal Waterways, Port and Coastal and Ocean Engineering Journal of ASCE. In press; initial-value computations of classical solitons.
Katsis, C. and Akylas, T. R.: 1987, Solitary waves in a rotating channel: A numerical study, Phys. Fluids 30(2), 297-301. Numerical solutions of RMKP Eq.
Keller, H. B.: 1977, Numerical solution of bifurcation and nonlinear eigenvalue problems, in P. Rabinowitz (ed.), Applications of Bifurcation Theory, Academic Press, New York, pp. 359-384.
References 553
Keller, H. B.: 1992, Numerical Methods for Two-Point Boundary- Value Problems, Dover, New York. Reprint of 1968 book plus 9 later journal articles.
Kessler, D. A. and Levine, H.: 1988, Pattern selection in three dimensional dendritic growth, Acta Metallurgica 36, 2693-2706.
Kessler, D. A., Koplik, J. and Levine, H.: 1988, Pattern selection in fingered growth phenomena, Adv. Phys. 37, 255-339.
Kevorkian, J. and Cole, J. D.: 1996, Multiple Scale and Singular Perturbation Methods, Springer-Verlag, New York.
Killingbeck, J.: 1977, Quantum-mechanical perturbation theory, Reports on Progress in Theoretical Physics 40, 977-1031. Divergence of asymptotic series.
Killingbeck, J.: 1978, A polynomial perturbation problem, Phys. Lett. A 67, 13-15.
Killingbeck, J.: 1981, Another attack on the sign-change argument, Chem. Phys. Lett. 80, 601-603.
Kindle, J.: 1983, On the generation of Rossby solitons during El Nino, in J. C. J. Nihoul (ed.), Hydrodynamics of the Equatorial Ocean, Elsevier, Amsterdam, pp. 353-368.
Kinsman, B.: 1965, Wind Waves, Prentice-Hall, Englewood Cliffs, New Jersey. Very readable treatise on water waves.
Kowalenko, V., Glasser, M. L., Taucher, T. and Frankel, N. E.: 1995, Generalised Euler-Jacobi inversion formula and asymptotics beyond all orders, Vol. 214 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge.
Kropinski, M. C. A., Ward, M. J. and Keller, J. B.: 1995, A hybrid asymptoticnumerical method for low Reynolds number flows past a cylindrical body, SIAM J. Appl. Math. 55(6), 1484-1510. Log-and-power series in Re.
Kruskal, M. D. and Segur, H.: 1985, Asymptotics beyond all orders in a model of crystal growth, Technical Report 85-25, Aeronautical Research Associates of Princeton.
Kruskal, M. D. and Segur, H.: 1991, Asymptotics beyond all orders in a model of crystal growth, Stud. Appl. Math. 85, 129-181.
554 WEAKLY NONLOCAL SOLITARY WAVES
Kubicek, M.: 1976, Algorithm 502: Dependence of solution of nonlinear systems on a parameter, ACM Trans. Math. Software 2, 98-107. Independent invention of pseudoarclength continuation for following a solution branch around a limit ("fold") point.
Kubota, T., Ko, R. S. and Dobbs, L. D.: 1978, Propagation of weakly nonlinear internal waves in a stratified fluid of finite depth, A. 1. A. A. J. Hydronautics 12, 157-165.
Kummer, M., Ellison, J. A. and Saenz, A. W.: 1991, Exponentially small phenomena in the rapidly forced pendulum, in H. Segur, S. Tanveer and H. Levine (eds), Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 197-211.
Laforgue, J. G. L. and O'Malley, Jr., R. E.: 1993, Supersensitive boundary value problems, in H. G. Kaper and M. Garbey (eds), Asymptotic and Numerical methods for Partial Differential Equations with Critical Parameters, Kluwer, Dordrecht, pp. 215-223.
Laforgue, J. G. L. and 0' Malley, Jr., R. E.: 1994, On the motion of viscous shocks and the supersensitivity of their steady-state limits, Methods and Applications of Analysis 1, 465-487. Exponential smallness in shock movement.
Laforgue, J. G. 1. and 0' Malley, Jr., R. E.: 1995a, Shock layer movement for Burgers' equation, SIAM J. Appl. Math. 55, 332-347.
Laforgue, J. G. L. and O'Malley, Jr., R. E.: 1995b, Viscous shock motion for advection-diffusion equations, Stud. Appl. Math. 95, 147-170.
Lanczos, C.: 1938, Trigonometric interpolation of empirical and analytical functions, Journal of Mathematics and Physics 17,123-199. The origin of both the pseudospectral method and the tau method. Lanczos is to spectral methods what Newton was to calculus.
Lange, C. G.: 1983, On spurious solutions of singular perturbation problems, Stud. Appl. Math. 68, 227-257. Shows that exponentially small terms resolves the spurious solutions found by power series for a class of nonlinear problems.
Lange, C. G. and Weinitschke, H. J.: 1994, Singular perturbations of elliptic problems on domains with small holes, Stud. Appl. Math. 92, 55-93. Log-and-power series for eigenvalues with comparisons with numerical solutions; demonstrates the surprisingly large sensitivity of eigenvalues to small holes in the membrane.
References 555
Larichev, V. D. and Reznik, G. M.: 1976, Two-dimensional Rossby soliton: An exact solution, Rep. USSR Acad. Sci. 231, 1077-1079. Discovery of Rossby modon.
Larsen, L. H.: 1965, Comments on 'solitary waves in the westerlies', J. Atmos. Sci. 22, 222-224. Rossby solitons.
Lazutkin, V. F., Schachmannski, I. G. and Tabanov, M. B.: 1989, Splitting of separatrices for standard and semistandard mappings, Physica D 40, 235-248.
Lee, J. and Ward, M. J.: 1995, On the asymtotic and numerical analyses of exponentially ill-conditioned singularly perturbed boundary value problems, Stud. Appl. Math. 94, 271-326. Eigenvalue problems and inhomogeneous boundary value problems for which an eigenvalue is exponentially small in 1/ € with applications to slow movements of shocks and boundary layer resonances.
Leonov, A. I.: 1981, The effect of Earth rotation in the propagation of weak nonlinear surface and internal long ocean waves, Annals of the New York Academy of Sciences 373, 150-159. RMKdV equation.
Leonov, A.I., Miropolsky, Y. Z. and Tamsalu, R. E.: 1979, Nonlinear stationary internal and surface waves in shallow seas, Tellus 31, 150-160.
Levine, H.: 1991, Dendritic crystal growth - overview, in H. Segur, S. Tanveer and H. Levine (eds) , Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 67-74.
Li, T.-Y.: 1987, Solving polynomial systems, Math. Intelligencer 9, 33-39. Probability-one homotopy.
Li, T.-Y.: 1997, Numerical solution of multivariate polynomial systems, in A. Iserles (ed.), Acta Numerica 1997, Acta Numerica, Cambridge University Press, New York. Review.
Li, T.-Y., Sauer, T. and Yorke, J. A.: 1988, Numerically determining solutions of systems of polynomial equations, Bulletin (New Series) of the American Mathematical Society 18(2), 173-177. Numerical; probability-one homotopy (continuation).
Lighthill, M. J.: 1958, An Introduction to Fourier Analysis and Generalized Functions, Cambridge University Press, New York. 80 pp.
Lim, R. and Berry, M. v.: 1991, Superadiabatic tracking for quantum evolution, J. Phys. A 24, 3255-3264.
556 WEAKLY NONLOCAL SOLITARY WAVES
Lin, Y. and Goff, RC.: 1988, A study of a mesoscale solitary wave in the atmosphere originating near a region of deep convection, J. Atmos. Sci. 45, 194-205. Nonlocal gravity solitons.
Liu, J. and Wood, A.: 1991, Matched asymptotics for a generalisation of a model equation for optical tunnelling, European J. Appl. Math. 2, 223-231. Compute the exponentially small imaginary part of the eigenvalue >., ~(>') rv exp( _1/E1/ n ), for the problem Uxx + (>. + EXn)U = 0 with outward radiating waves on the semi-infinite interval.
Liu, Y., Liu, L. and Tang, T.: 1994, The numerical computation of connecting orbits in dynamical systems: A rational spectral approach, J. Comput. Phys. 111, 373-380.
Lombardi, E.: 1996, Homoclinic orbits to small periodic orbits for a class of reversible systems, Proceedings of the Royal Society of Edinburgh 126, 1035-1054. "Homoclinic-to-periodic" is a synonym for nonlocal solitary wave; rigorous analysis of a family of systems that includes the FKdV equation.
Long, R R: 1953, Some aspects of the flow of stratified fluids, Pt. 1: Theoretical investigation, Tellus 5, 42-57. Independent derivation of the DubreilJacotin-Long equation for nonlinear internal gravity waves.
Long, R R: 1956, Solitary waves in one- and two-fluid systems, Tellus 8,460-471. Classical gravity solitons.
Long, R R: 1964, Solitary waves in the westerlies, J. Atmos. Sci. 21, 197-200. Classical solitons.
Long, R R: 1965, On the Boussinesq approximation and its role in the theory of internal waves, Tellus 17, 46-52. Classical solitons.
Long, R R. and Morton, J. B.: 1966, Solitary waves in compressible stratified fluids, Tellus 18, 79-85.
Lorenz, E. N. and Krishnamurthy, V.: 1987, On the nonexistence of a slow manifold, J. Atmos. Sci. 44, 2940-2950. Weakly non-local in time.
Lozano, C. and Meyer, RE.: 1976, Leakage and response of waves trapped by round islands, Phys. Fluids 19, 1075-1088. Leakage is exponentially small in the perturbation parameter.
Lozier, D. W. and Olver, F. W. J.: 1994, Numerical evaluation of special functions, in W. Gautschi (ed.), Math. Comp. 1943-1993: A Half-Century of Computational Mathematics, number 48 in Proceedings of Symposia in Applied Mathematics, American Mathematical Society, American Mathematical Society, Providence, Rhode Island. Hyperasymptotics.
References 557
Lu, c., MacGillivray, A. D. and Hastings, S. P.: 1992, Asymptotic behavior of solutions of a similarity equation for laminar flows in channels with porous walls, [MA J. Appl. Math. 49, 139-162. Beyond-aIl-orders matched asymptotics.
Lund, J. and Bowers, K. L.: 1992, Sinc Methods For Quadrature and Differential Equations, Society for Industrial and Applied Mathematics, Philadelphia. 304 pp. Numerical.
Lynch, P.: 1992, Richardson's barotropic forecast: A reappraisal, Bull. Amer. Meteor. Soc. 73(1), 35-47.
Lyness, J. N. and Ninham, B. W.: 1967, Numerical quadrature and asymptotic expansions, Math. Comp. 21, 162-. Shows that the error is a power series in the grid spacing h plus an integral which is transcendentally small in Ilh.
MacGillivray, A. D.: 1997, A method for incorporating transcendentally small terms into the method of matched asymptotic expansions, Stud. Appl. Math. 99, 285-310. Linear example has a general solution which is the sum of an antisymmetric function A(x) which is well-approximated by a second-derivative-dropping outer expansion plus a symmetric part which is exponentially small except at the boundaries, and can be approximated only by a WKB method (with second derivative retained). His nonlinear example is the Carrier-Pearson problem whose exact solution is a KdV cnoidal wave, but required to satisfy Dirichlet boundary conditions. Matching fails because each soliton peak can be translated with only an exponentially small error; MacGillivray shows that the peaks, however many are fit between the boundaries, must be evenly spaced.
MacGillivray, A. D. and Lu, c.: 1994, Asymptotic solution of a laminar flow in a porous channel with large suction: A non linear turning point problem, Methods and Applications of Analysis 1, 229-248. Incorporation of exponentially small terms into matched asymptotics.
MacGillivray, A. D., Liu, B. and Kazarinoff, N. D.: 1994, Asymptotic analysis of the peeling-off point of a French duck, Methods and Applications of Analysis 1, 488-509. Beyond-all-orders theory.
Mackay, R. S.: 1991, Exponentially small residues near analytic invariant circles, in H. Segur, S. Tanveer and H. Levine (eds), Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 365-374.
558 WEAKLY NONLOCAL SOLITARY WAVES
Mahapatra, P. R., Doviak, R. J. and Zrnic, D. S.: 1991, Multisensor observation of an atmospheric undular bore, Bull. Amer. Meteor. Soc. 72(10), 1468-1480. Nonlocal horizontally-ducted internal gravity waves, visualized by satellite photographs and false-color radar imagery.
Malanotte Rizzoli, P.: 1982, Planetary solitary waves in geophysical flows, Advances in Geophysics 24, 147-224. Review; Rossby waves.
Mallier, R.: 1995, Stuart vortices on a beta-plane, Dyn. Atmos. Oceans 22, 213-238. Classical Rossby solitary waves.
Malomed, B. A.: 1993, Bound states of envelope solitons., Phys. Rev. E 47, 2784-. TNLS Eq.
Marshall, H. G. and Boyd, J. P.: 1987, Solitons in a continuously stratified equatorial ocean, J. Phys. Oceangr. 17, 1016-1031. [Hermite functions; coupling between different vertical modes].
Martin, O. and Branis, S. V.: 1991, Solitary waves in self-induced transparency, in H. Segur, S. Tanveer and H. Levine (eds) , Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 327-336.
Maslov, V. P.: 1994, The Complex WKB Method for Nonlinear Equations I: Linear Theory, Birkhauser, Boston. Calculation of exponentially small terms.
Maslowe, S. A. and Redekopp, L. G.: 1979, Solitary waves in stratified shear flows, Geophys. Astrophys. Fluid Dyn. 13, 185-196. Gravity wave nonlocal solitons, radiating upward from a leaky waveguide.
Maslowe, S. A. and Redekopp, L. G.: 1980, Long nonlinear waves in stratified shear flows, J. Fluid Mech. 101, 321-348.
Masuda, A.: 1988, A skewed eddy of Batchelor-Modon type, Journal of the Oceanographic Society of Japan 44, 189-199. Classical modon.
Maxworthy, T.: 1980, On the formation of nonlinear internal waves from the gravitational collapse of mixed regions in two and three dimensions, J. Fluid Mech. 96, 47-64. Laboratory experiments on the generation of internal gravity wave solitons.
Maxworthy, T.: 1983, Experiment on solitary internal Kelvin waves, J. Fluid Mech. 129, 365-383.
Maxworthy, T. and Redekopp, L. G.: 1976a, New theory of the Great Red Spot from solitary waves in the Jovian atmosphere, Nature 260,509-511.
References 559
Maxworthy, T. and Redekopp, L. G.: 1976b, A solitary wave theory of the Great Red Spot and other observed features in the Jovian atmosphere, Icarus 29, 261-271.
Maxworthy, T., Redekopp, L. G. and Weidman, P. D.: 1978, On the production and interaction of planetary solitary waves: Applications to the Jovian atmosphere, Icarus 33, 388-409.
McArthur, K. M., Bowers, K. L. and Lund, J.: 1987, Numerical implementation of the sinc-Galerkin method for second-order hyperbolic equations, Numer. Meth. Partial Differential Eq. 3, 169-185. Numerical; not soliton.
McLeod, J. B.: 1991, Laminar flow in a porous channel, in H. Segur, S. Tanveer and H. Levine (eds) , Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 255-266.
McLeod, J. B.: 1992, Smoothing of Stokes discontinuities, Proc. Roy. Soc. London, Series A 437, 343-354.
McWilliams, J. C.: 1991, Geostrophic vortices, in A. R. Osborne (ed.), Nonlinear Topics of Ocean Physics: Fermi Summer School, Course LIX, NorthHolland, Amsterdam, pp. 5-50.
McWilliams, J. C. and Flierl, G. R.: 1979, On the evolution of isolated nonlinear vortices, J. Phys. Oceangr. 9, 1155-1182.
Meiss, J. D. and Horton, W.: 1983, Solitary drift waves in the presence of magnetic shear, Phys. Fluids 26,990-997. Show that plasma modons leak radiation for large lxi, and therefore are nanopterons.
Meleshko, V. V. and van Heijst, G. J. F.: 1992, On Chaplygin's investigations of two-dimensional vortex structures in an inviscid fluid, J. Fluid Mech. 272, 157-182. Very good history of the original discovery of modons by Chapyglin and Lamb around 1902; review with some original analysis.
Melville, W. K., Tomasson, G. G. and Renouard, D. P.: 1989, On the stability of Kelvin waves, J. Fluid Mech. 206, 1-23. Numerical solutions for barotropic gravity waves in shallow water with rotation, a system slightly more complex than the RMKP equation. Show that the Kelvin solitary wave develops a curved front and loses energy because of resonant coupling with the Poincare waves, which are the oscillatory tail for these nonlocal solitons.
Menhofer, A., Smith, R. K., Reeder, M. J. and Christie, D. R.: 1997, 'morning glory' disturbances and the environment in which they propagate, J.
560 WEAKLY NONLOCAL SOLITARY WAVES
Atmos. Sci. 54, 1712-1725. Argue that radiative leakage from the waveguide is equally strong both on days when Morning Glory solitary waves are observed and on days when they are not, so continuous forcing of the Morning Glory by mesoscale circulations is essential to soliton longevity.
Merzbacher, E.: 1970, Quantum Mechanics, 2 edn, WHey, New York. 400 pp. Hermite funcs.; Galerkin meths. in quantum mechanics.
Meyer, R. E.: 1973a, Adiabatic variation. Part 1: Exponential property for the simple oscillator, J. Appl. Math. Phys. ZAMP 24, 517-524.
Meyer, R. E.: 1973b, Adiabatic variation. Part II: Action change for simple oscillator, J. Appl. Math. Phys. ZAMP.
Meyer, R. E.: 1974a, Exponential action of a pendulum, Bull. Amer. Math. Soc. 80, 164-168.
Meyer, R. E.: 1974b, Adiabatic variation. Part IV: Action change of a pendulum for general frequency, J. Appl. Math. Phys. ZAMP 25, 651-654.
Meyer, R. E.: 1975, Gradual reflection of short waves, SIAM J. Appl. Math. 29, 481-492.
Meyer, R. E.: 1976a, Adiabatic variation. Part V: Nonlinear near-periodic oscillator, J. Appl. Math. and Physics ZAMP 27, 181-195.
Meyer, R. E.: 1976b, Quasiclassical scattering above barriers in one dimension, J. Math. Phys. 17, 1039-1041.
Meyer, R. E.: 1979, Surface wave reflection by underwater ridges, J. Phys. Oceangr. 9, 150-157.
Meyer, R. E.: 1980, Exponential asymptotics, SIAM Rev. 22, 213-224.
Meyer, R. E.: 1982, Wave reflection and quasiresonance, Theory and Application of Singular Perturbation, number 942 in Lecture Notes in Mathematics, Springer-Verlag, New York, pp. 84-112.
Meyer, R. E.: 1986, Quasiresonance of long life, J. Math. Phys. 27, 238-248.
Meyer, R. E.: 1989, A simple explanation of Stokes phenomenon, SIAM Rev. 31, 435-444.
Meyer, R. E.: 1990, Observable tunneling in several dimensions, in R. Wong (ed.), Asymptotic and Computational Analysis, Marcel Dekker, New York, pp. 299-328.
References 561
Meyer, R. E.: 1991a, On exponential asymptotics for nonseparable wave equations 1: Complex geometrical optics and connection, SIAM J. Appl. Math. 51, 1585-1601.
Meyer, R. E.: 1991b, On exponential asymptotics for nonseparable wave equations I: EBK quantization, SIAM J. Appl. Math. 51, 1602-1615.
Meyer, R. E.: 1991c, Exponential asymptotics for partial differential equations, in H. Segur, S. Tanveer and H. Levine (eds) , Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 29-36.
Meyer, R. E.: 1992, Approximation and asymptotics, in D. A. Martin and G. R. Wickham (eds) , Wave Asymptotics, Cambridge University Press, New York, pp. 43-53. Blunt and perceptive review.
Meyer, R. E. and Guay, E. J.: 1974, Adiabatic variation. Part Ill: A deep mirror model, J. Appl. Math. and Physics ZAMP 25, 643-650.
Meyer, R. E. and Painter, J. F.: 1979, Wave trapping with shore absorption, J. Engineering Math. 13,33-45.
Meyer, R. E. and Painter, J. F.: 1981, New connection method across more general turning points, Bull. Amer. Math. Soc. 4, 335-338.
Meyer, R. E. and Painter, J. F.: 1982, Irregular points of modulation, Adv. Appl. Math. 4, 145-174.
Meyer, R. E. and Painter, J. F.: 1983a, Connection for wave modulation, SIAM J. Math. Anal. 14, 450-462.
Meyer, R. E. and Painter, J. F.: 1983b, On the Schroedinger connection, Bull. Amer. Math. Soc. 8, 73-76.
Meyer, R. E. and Shen, M. C.: 1991, On Floquet's theorem for nonseparable partial differential equations, in B. D. Sleeman (ed.), Eleventh Dundee Conference in Ordinary and Partial Differential Equations, Pitman Advanced mathematical Research Notes, Longman-Wiley, New York, pp. 146-167.
Meyer, R. E. and Shen, M. C.: 1992, On exponential asymptotics for nonseparable wave equations Ill: Approximate spectral bands of periodic potentials on strips, SIAM J. Appl. Math. 52, 730-745.
Mied, R. P. and Lindemann, G. J.: 1979, The propagation and evolution of cyclonic Gulf Stream Rings, J. Phys. Oceangr. 9, 1183-1206. Radiative decay of ocean vortices.
562 WEAKLY NONLOCAL SOLITARY WAVES
Mied, R. P. and Lindemann, G. J.: 1982, The birth and evolution of eastward propagating modons, J. Phys. Oceangr. 12, 213-230. Radiative decay of ocean vortices.
Miesen, R. H. M., Kamp, L. P. J. and Sluijter, F. W.: 1990a, Long solitary waves in compressible shallow fluids, Phys. Fluids A 2, 359-370. Internal gravity solitons.
Miesen, R. H. M., Kamp, L. P. J. and Sluijter, F. W.: 1990b, Long solitary waves in compressible deep fluids, Phys. Fluids A 2, 1401-1411. Internal gravity solitons.
Miles, J. W.: 1979, On solitary Rossby1 waves, J. Atmos. Sci. 36, 1236-1238. KdV with mixed cubic/quadratic nonlinearity; does not discuss nonlocal solitons.
Miles, J. W.: 1980, Solitary waves, Ann. Reviews Fluid Mech. 12, 11-43.
Milewski, P.: 1993, Oscillating tails in the perturbed Korteweg-deVries equation, Stud. Appl. Math. 90, 87-90. FKdV Eq.
Miller, G. F.: 1966, On the convergence of Chebyshev series for functions possessing a singularity in the range of representation, SIAM J. Numer. Anal. 3, 390-409.
Mills, R. D.: 1987, Using a small algebraic manipulation system to solve differential and integral equations by variational and approximation techniques, J. Symbolic Comp. 3, 291-301.
Moore, D. W. and Philander, S. G. H.: 1977, Modelling of the tropical oceanic circulation, in E. D. Goldberg (ed.), The Sea, number 6, Wiley, New York, pp. 319-361. Review of tropical oceanography.
Morbidelli, A. and Giorgilli, A.: 1995, Superexponential stability of KAM tori, J. Statistical Phys. 78, 1607-1617. Estimates of the exponentially long time scale for Arnold diffusion.
Morgan, A. and Sommese, A.: 1987, Computing all solutions to a polynomial systems using homotopy continuation, Appl. Math. Comput. 24, 115-138.
Morgan, A. P.: 1987, Solving Polynomial Systems Using Continuation for Scientific and Engineering Problems, Prentice-Hall, Englewood Cliffs, New Jersey.
Morse, P. M. and Feshbach, H.: 1953, Methods of Theoretical Physics, McGrawHill, New York. 2000 pp, (in two volumes).
References 563
Mulholland, L. S. and Sloan, D. M.: 1992, The role of preconditioning in the solution of evolutionary partial differential equations by implicit Fourier pseudospectral methods, J. Comput. Appl. Math. 42, 157-174. Timeintegration of a wave equation with Fourier spatial discretization.
Nagashima, H. and Kuwahara, M.: 1981, J. Phys. Soc. 50, 3792. Soliton of fifth-degree Korteweg-de Vries equation.
Nayfeh, A. H.: 1973, Perturbation Methods, Wiley, New York. Good reference on the method of multiple scales.
Neal, A. B., Butterworth, I. J. and Murphy, K. M.: 1977, The morning glory, Weather 32, 176-183. Semi-popular review with photos.
Nekhoroshev, N. N.: 1977, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems, Russ. Math. Surveys 32, 1-65. Estimates of the exponentially small time scale for Arnold diffusion.
Neven, E. C.: 1991, Quadrupole modons on a sphere, in G. J. F. van Heijst (ed.), Coherent Vortex Structures, Instituut voor Meteorologie en Oceanografie (IMOU) for Rijksuniversiteit Utrecht, Utrecht.
Neven, E. C.: 1992, Quadrupole modons on a sphere, Geophys. Astrophys. Fluid Dyn. 65, 105-126. Nonlocal modons.
Neven, E. C.: 1994a, Baroclinic modons on a sphere, J. Atmos. Sci. 51, 1447-1464.
Neven, E. C.: 1994b, Determination of the linear stability of modons on a sphere by high-truncation time integrations, in G. J. F. van Heijst (ed.), Modelling of Ocean Vortices, North-Holland, Amsterdam.
Neven, E. C.: 1994c, Modons in shear flow on a sphere, Geophys. Astrophys. Fluid Dyn. 74,51-71.
Nezlin, M. V. and Snezhkin, E. N.: 1993, Rossby Vortices, Spiral Structures, Solitons, Springer-Verlag, New York.
Nihoul, J. C. J. and Jamart, B. M. (eds): 1989, International Liege Colloquium on Ocean Hydrodynamics, number 20 in Liege Colloquium on Ocean Hydrodynamics, Elsevier, Amsterdam.
Noonan, J. A. and Smith, R. K.: 1985, Linear and weakly nonlinear internal wave theories applied "morning glory" waves, Geophys. Astrophys. Fluid Dyn. 33, 123-143.
564 WEAKLY NONLOCAL SOLITARY WAVES
Noonan, J. A. and Smith, R. K.: 1986, Sea breeze circulations over Cape York Peninsula and the generation of the Gulf of Carpentaria cloud line disturbances, J. Atmos. Sci. 43, 1679-1693. Nonlocal gravity solitons (Morning Glory).
Noonan, J. A. and Smith, R. K.: 1987, The generation of north Australian cloud lines and the "morning glory", Austral. Meteor. Mag. 35, 31-45. Weakly nonlocal, horizontally-ducted solitary waves and the role of mesoscale circulations in triggering them.
Novokshenov, V. Y.: 1993, Nonlinear Stokes Phenomenon for the second Painleve equation, Physica D 63, 1-7.
Nycander, J.: 1988, New stationary vortex solutions of the Hasegawa-Mima equation, J. Plasma Phys. 39, 413-430. Hasegawa-Mima is another name for the quasi-geostrophic equation; analytical and perturbative solutions.
Nycander, J. and Sutyrin, G. G.: 1992, Steadily translating anticyclones on the beta plane, Dyn. Atmos. Oceans 16, 473-498. Approximate analytical solutions for almost axisymmetric anticyclonic vortices; these are weakly nonlocal through Rossby radiation emitted from the equatorial flank of the vortex.
Oberhettinger, F.: 1973, Fourier Expansions: A Collection of Formulas, Academic Press, New York.
Oikawa, M.: 1993, Effects of third order dispersion on the Nonlinear Schroedinger equation, J. Phys. Soc. Japan 62, 2324-2333.
Olde Daalhuis, A. B.: 1992, Hyperasymptotic expansions of confluent hypergeometric functions, [MA J. Appl. Math. 49, 203-216.
Olde Daalhuis, A. B.: 1993, Hyperasymptotics and the Stokes phenomenon, Proc. Roy. Math. Soc. Edinburgh A 123, 731-743.
Olde Daalhuis, A. B.: 1995, Hyperasymptotic solutions of second-order linear differential equations 11, Methods of Applicable Analyis 2, 198-211.
Olde Daalhuis, A. B.: 1996, Hyperterminants 11, J. Comput. Appl. Math. 76, 255-264. Convergent series for the generalized Stieltjes functions that appear in hyperasymptotic expansions.
Olde Daalhuis, A. B. and Olver, F. W. J.: 1994, Exponentially improved asymptotic solutions of ordinary differential equations. 11. Irregular singularities of rank one, Proc. Roy. Soc. London A 445, 39-56.
References 565
Olde Daalhuis, A. B. and Olver, F. w. J.: 1995a, Hyperasymptotic solutions of second-order linear differential equations. I, Methods of Applicable Analysis 2, 173-197.
Olde Daalhuis, A. B. and Olver, F. W. J.: 1995b, On the calculation of Stokes multipliers for linear second-order differential equations, Methods of Applicable Analysis 2, 348-367.
Olde Daalhuis, A. B. and Olver, F. W. J.: 1995c, Exponentially-improved asymptotic solutions of ordinary differential equations. II: Irregular singularities of rank one, Proc. Roy. Soc. London A 2, 39-56.
Olde Daalhuis, A. B., Chapman, S. J., King, J. R., Ockendon, J. R. and Tew, R. H.: 1995, Stokes phenomenon and matched asymptotic expansions, SIAM J. Appl. Math. 6, 1469-1483.
Olson, D. B.: 1991, Rings in the ocean, Annual Reviews of Earth and Planetary Science 38, 283-311. Observations of ocean vortices.
Olver, F. W. J.: 1974, Asymptotics and Special Functions, Academic, New York.
Olver, F. W. J.: 1990, On Stokes' phenomenon and converging factors, in R. Wong (ed.), Asymptotic and Computational Analysis, Marcel Dekker, New York, pp. 329-356.
Olver, F. W. J.: 1991a, Uniform, exponentially-improved asymptotic expansions for the generalized exponential integral, SIAM J. Math. Anal. 22,1460-1474.
Olver, F. W. J.: 1991b, Uniform, exponentially-improved asymptotic expansions for the confluent hypergeometric function and other integral transforms, SIAM J. Math. Anal. 22, 1475-1489.
Olver, F. W. J.: 1993, Exponentially-improved asymptotic solutions of ordinary differential equations I: The confluent hypergeometric function, SIAM J. Math. Anal. 24, 756-767.
Olver, F. W. J.: 1994, Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations, Methods of Applicable Analysis 1(1), 1-13.
Olver, P. J.: 1986, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York.
566 WEAKLY NONLOCAL SOLITARY WAVES
O'Malley, Jr., R. E.: 1991, Singular perturbations, asymptotic evaluation of integrals, and computational challenges, in H. G. Kaper and M. Garbey (eds), Asymptotic Analysis and the Numerical Solution of Partial Differential Equation, Dekker, New York, pp. 3-16.
Oppenheimer, J. R.: 1928, Three notes on the quantum theory of aperiodic effects, Phys. Rev. 31, 66-81. Shows that Zeeman effect generates an imaginary part to the energy which is exponentially small in the reciprocal of the perturbation parameter.
Ortega, J. M. and Rheinboldt, W. C.: 1970, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York.
Ostrovsky, L. A.: 1978, Nonlinear internal waves in a rotation ocean, Oceanology 18, 181-191. First derivation of the Rotation-Modified Kortewegde Vries equation (RMKdV); micropterons in one space dimension.
Ostrovsky, L. A. and Stepanyants, Y. A.: 1990, Nonlinear surface and internal waves in rotating fluids, in A. V. Gaponov-Grekhov, M. 1. Rabinovich and J. Engelbrecht (eds), Nonlinear Waves 3, Springer, New York, pp. 106-128. RMKdV eq.
Painter, J. F. and Meyer, R. E.: 1982, Turning-point connection at close quarters, SIAM J. Math. Anal. 13,541-554.
Paris, R. B.: 1992a, Smoothing of the Stokes phenomenon for high-order differential equations, Proc. Roy. Soc. London A 436, 165-186.
Paris, R. B.: 1992b, Smoothing of the Stokes phenomenon using Mellin-Barnes integrals, J. Comput. Appl. Math. 41, 117-133.
Paris, R. B. and Wood, A. D.: 1989, A model for optical tunneling, IMA J. Appl. Math. 43, 273-284. Exponentially small leakage from the fiber.
Paris, R. B. and Wood, A. D.: 1992, Exponentially-improved asymptotics for the gamma function, J. Comput. Appl. Math. 41, 135-143.
Paris, R. B. and Wood, A. D.: 1995, Stokes phenomenon demystified, IMA Bulletin 31, 21-28. Short review of hyperasymptotics.
Pasmanter, R. A.: 1994, On long lived vortices in 2-D viscous flows, most probable states of inviscid 2-D flows and a soliton equation, Phys. Fluids A 6, 1236-1241. Sinh-Poisson equation.
Pereira, N. R. and Redekopp, L. G.: 1980, Radiation damping of long, finiteamplitude internal waves, Phys. Fluids 23, 2182-2183.
References 567
Petviashvili, V. I.: 1976, Equation of an extraordinary soliton, Soviet J. Plasma Phys. 2, 257-260. Interesting non-Newtonian iteration, applied to compute a classical soliton.
Petviashvili, V. I.: 1981, Red Spot of Jupiter and the drift soliton in a plasma, Soviet Physics JETP Lett. 32, 619-622. Flierl-Petviashvili classical axisymmetric soliton.
Petviashvili, V. I. and Tsvelodub, O. Y.: 1978, Horshoe-shaped solitons on a flowing viscous film of fluid, Soviet Phys. Doklady 23, 117-118. NonNewtonian iteration, classical soliton.
Pham, F.: 1988, Resurgence, quantized canonical transformations and multiinstanton expansions, Algebraic Analysis, Vol. 2, Academic Press, New York, pp. 699-726. Beyond all orders perturbation theory.
Physick, W.: 1987, Observations of the solitary wave train at Melbourne, Australia, Austral. Meteor. Mag. 34, 163-172. Nonlocal internal gravity waves.
Pokrovskii, V. L.: 1989, in I. M. Khalatnikov (ed.), Landau, the Physicist and the Man: Recollections of L. D. Landau, Pergamon Press, Oxford. Relates the amusing story that the Nobel Laureate Lev Landau believed the Prokrovskii-Khalatnikov( 1961) "beyond-all-orders" method was wrong. The correct answer (but with incorrect derivation) is given in the LandauLifschiftz textbooks. Eventually, Landau realized his mistake and apologized.
Pokrovskii, V. L. and Khalatnikov, I. M.: 1961, On the problem of abovebarrier reflection of high-energy particles, Soviet Phys. JETP 13, 1207-1210. Applies matched asymptotics in the complex plane to compute the exponentially small reflection which is missed by WKB.
Pomeau, Y.: 1991, Singular perturbations of solitons, in H. Segur, S. Tanveer and H. Levine (eds), Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 241-246.
Pomeau, Y., Ramani, A. and Grammaticos, G.: 1988, Structural stability of the Korteweg-deVries solitons under a singular perturbation, Physica D 21, 127-134. Weakly nonlocal solitons of the FKdV equation; complexplane matched asymptotics.
Pomeranz, S. B.: 1992, A solitary wave application of an iterative method for nonlinear elliptic eigenvalue problems, Computer Mathematics and Applications 23, 45-49.
568 WEAKLY NONLOCAL SOLITARY WAVES
Praddaude, H. C.: 1972, Energy levels of hydrogen like atoms in a magnetic field, Phys. Rev. A 6, 1321-1324.
Press, W. H., Flannery, B. H., Teukolsky, S. A. and Vetterling, W. T.: 1986, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, New York.
Proudman, I. and Pearson, J. RA.: 1957, Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder, J. Fluid Mech. 2, 237-262. Log-and-powers expansion.
Raithby, G.: 1971, Laminar heat transfer in the thermal entrance region of circular tubes and two-dimensional rectangular ducts with wall suction and injection, Internat. J. Heat Mass Transfer 14, 223-243.
Rakovic, M. J. and Solov'ev, E. A.: 1989, Higher orders of semiclassical expansion for the one-dimensional Schroedinger equation, Phys. Rev. A 40, 6692-6694.
Ramamurthy, M. K., Collins, B. P., Rauber, R M. and Kennedy, P. C.: 1990, Evidence of very-large-amplitude solitary waves in the atmosphere, Nature 348, 314-317. Vertical displacements as large as 4 km; these solitons are weakly nonlocal through radiative leakage to the upper atmosphere.
Reddy, S. C., Schmid, P. J. and Henningson, D. S.: 19, Pseudospectra of the Orr-Sommerfeld equation, SIAM J. Appl. Math. 53, 15-47. Exponentially sensitive eigenvalues.
Redekopp, L. G.: 1977, On the theory of solitary Rossby waves, J. Fluid Mech. 82, 725-745. Classical solitons in mean shear.
Redekopp, L. G.: 1990, Nonlinear internal waves in geophysics: Long internal waves, Lectures in Applied Mathematics 20,59-78. Good review of solitary waves with discussion of radiative leakage of ducted gravity waves and a derivation of the RMKdV Eq.
Redekopp, L. G. and Weidman, P. D.: 1978, Solitary Rossby waves in zonal shear flows and their interactions, J. Atmos. Sci. 35, 790-804.
Reeder, M. J., Christie, D. R, Smith, R K. and Grimshaw, R: 1995, Interacting "morning glories" over Northern Australia, Bull. Amer. Meteor. Soc. 76(7),1165-1171. Review combined with a detailed analysis of colliding Morning Glories; spectacular satellite photos.
References 569
Reinhardt, W. P.: 1982, Pade summation for the real and imaginary parts of atomic Stark eigenvalues, Int. J. Quantum Chem. 21, 133-146. Two successive Pade transformations are used to compute the exponentially small imaginary part of the eigenvalue.
Renouard, D. P., Chabert d'Hieres, G. and Zhang, x.: 1987, An experimental study of strongly nonlinear waves in a rotating system, J. Fluid Mech. 177, 381-394. These experiments on internal solitary waves in a rotating channel inspired much later theory (RMKP Eq.).
Reyna, L. G. and Ward, M. J.: 1994, Resolving weak internal layer interactions for the Ginzburg-Landau equation, European J. Appl. Math. 5, 495-523. Introduces a nonlinear WKB-type transformation which magnifies exponentially weak boundary layers to give a well-conditioned transformed problem. Applications to Allen-Cahn equation and double-well nonlinearities.
Reyna, L. G. and Ward, M. J.: 1995a, On the exponentially slow motion of a viscous shock, Co mm. Pure Appl. Math. 48, 79-120. Slow-moving shocks for Ut + [J(u)lx = fUxx where the nonlinearity f(u) is symmetric in u and convex; a special case is Burgers' equation. Beyond-aIl-orders asymptotics plus numerical solutions which use a WKB-type transformation of the problem to a well-conditioned form.
Reyna, L. G. and Ward, M. J.: 1995b, Metastable internal layer dynamics for the viscous Cahn-Hilliard equation, Methods and Applications of Analysis 2(3), 285-306. Beyond-all-orders-asymptotics is compared with numerical solutions for the exponentially slow motion of internal layers.
Reyna, L. G. and Ward, M. J.: 1995c, On exponential ill-conditioning and internallayer behavior, Journal of Numerical Functional Analysis and Optimization.
Reznik, G. M., Grimshaw, R. H. J. and Sriskandarajah, K.: 1997, On basic mechanisms governing the evolution of two-layer localized quasigeostrophic vortices on a ,a-plane., Geophysical Astrophysical Fluid Dynamics 86, 1-42. Includes radiative decay.
Richardson, L. F.: 1910, The approximate arithmetical solution by finite differences of physical problems involving differential equations with an application to stresses in a masonry dam, Phil. Trans. Royal Society of London A 210, 307-357. Invention of Richardson's iteration.
Richardson, L. F.: 1922, Weather Prediction by Numerical Processes, Cambridge University Press, New York.
570 WEAKLY NONLOCAL SOLITARY WAVES
Richardson, L. F.: 1927, The deferred approach to the limit. Part 1.- Single lattice, Phil. Trans. Royal Soc. 226, 299-349. Invention of Richardson extrapolation, which is an asymptotic but divergent procedure because of beyond-all-orders terms in the grid spacing h. Reprinted in Richardson's Collected Papers, ed. by O. M. Ashford et al.
Richardson, P. L.: 1993, Tracking ocean eddies, American Scientist 81(3),261-271. Good review with many excellent diagrams of Gulf Stream rings, meddies, retroflection eddies and other ocean vortices.
Robinson, W. A.: 1976, The existence of multiple solutions for the laminar flow in a uniformly porous channel with suction at both walls, J. Engineering Math. 10,23-40. Exponentially small difference between two distinct nonlinear solutions.
Rosser, J. B.: 1951, Transformations to speed the convergence of series, Journal of Research of the National Bureau of Standards 46, 56-64. Convergence factors; improvements to asymptotic series.
Rosser, J. B.: 1955, Explicit remainder terms for some asymptotic series, Journal of Rational Mechanics and Analysis 4, 595-626.
Rottman, J. W. and Einaudi, F.: 1993, Solitary waves in the atmosphere, J. Atmos. Sci. 50(14), 2116-2136. Good review of theory; two theorywith-observations case studies: one of a Morning Glory and the other of a convection-triggered solitary wave (horizontally-ducted internal gravity weakly nonlocal solitons.
Rottman, J. W. and Simpson, J. E.: 1989, The formation of internal bores in the atmosphere: A laboratory model, Quart. J. Royal Meteor. Soc. 115, 941-963. Laboratory experiments relevant to the Morning Glory.
Russell, J. S.: 1838, Seventh report of the committee on waves, Transactions of the British Association for the Advancement of Science 6, 417-496. An early report on solitary waves by the man who discovered them, and later designed the largest ship of the nineteenth century, the Great Eastern.
Sanders, B. F., Katopodes, N. and Boyd, J. P.: 1998, Spectral modeling of nonlinear dispersive waves, J. Hydraulic Engineering of ASCE 124(1), 2-12. Numerical calculations of classical solitary waves.
Scales, L. E. and Wait, R.: 1981, Methods for systems of algebraic equations, in C. T. H. Baker and C. Phillips (eds), The Numerical Solution of Nonlinear Problems, Clarendon Press (Oxford University Press), Oxford, pp. 20-31.
References 571
Scheurle, J., Marsden, J. E. and Holmes, P.: 1991, Exponentially small estimate for separatrix splitting, in H. Segur, S. Tanveer and H. Levine (eds), Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 187-196. Show that the splitting is proportional to v( f) exp( -7r / (2f)) where v( epsilon) has as essential singularity at f = 0 and must be represented as a Laurent series rather than a power series. No examples of essentially-singular V(f) for nonlocal solitons are as yet known.
Schraiman, B. I.: 1986, On velocity selection and the Saffman-Taylor problem, Phys. Rev. Letters 56, 2028-2031.
Schulten, Z., Anderson, D. G. M. and Gordon, R. G.: 1979, An algorithm for the evaluation of the complex Airy functions, J. Comput. Phys. 31,60-75. An alternative to hyperasymptotics - a very efficient one.
Segur, H.: 1991, The geometric model of crystal growth, in H. Segur, S. Tanveer and H. Levine (eds), Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 29-36.
Segur, H. and Kruskal, M. D.: 1987, On the nonexistence of small amplitude breather solutions in 4J4 theory, Phys. Rev. Letters 58, 747-750. Title not withstanding, the 4J4 breather does exist, but is nonlocal.
Segur, H., Tanveer, S. and Levine, H. (eds): 1991, Asymptotics Beyond All Orders, Plenum, New York. 389pp.
Seydel, R.: 1988, From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis, Elsevier, Amsterdam. 355 pp.; a second edition has since appeared. Nothing about solitary waves, but good undergraduate introduction to solving systems of nonlinear equations and continuation.
Shen, C. y.: 1981, On the dynamics of a solitary vortex, Dyn. Atmos. Oceans 5, 239--267. Nonexistence of Rossby monopole vortices as classical solitons.
Shen, M. C. and Sun, S. M.: 1991, Generalized solitary waves in a stratified fluid, in H. Segur, S. Tanveer and H. Levine (eds) , Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 299-307.
Shepherd, T. G.: 1990a, A general method for finding extremal energy states of Hamiltonian dynamical systems, with applications to perfect fluid, J. Fluid Mech. 213, 573-587. Numerical method for computing solitary waves and other steady or steadily-translating nonlinear structures through an enstrophy-preserving Hamiltonian damping.
572 WEAKLY NONLOCAL SOLITARY WAVES
Shepherd, T. G.: 1990b, Symmetries, conservation laws, and Hamiltonian structure in geophys. fluid dyn., in R. Dmowska and B. Saltzman (eds), Advances in Geophysics 32, Academic Press, Boston, pp. 287-338. Derives Hamiltonian formulations and Casimirs for a wide variety of fluid models including the shallow water wave equations with rotation, threedimensional compressible flow, etc. Brief discussion of how these can be used to compute nonlinear coherent structures by means of an enstrophypreserving dissipation (or anti-dissipation).
Sibuya, Y.: 1990, Gevrey property of formal solutions in a parameter, in R. Wong (ed.), Asymptotic and Computational Analysis, Marcel Dekker, New York, pp. 393-401.
Simpson, J. E.: 1987, Gravity Currents - in the Environment and the Laboratory, Ellis Horwood, Chichester, West Sussex, England. 244 pp.
Skinner, L. A.: 1975, Generalized expansions for slow flow past a cylinder, Quart. J. Mech. Appl. Math. 28, 333-340. Log-and-power-series in Re.
Smith IV, D. C. and Reid, R. 0.: 1982, A numerical study of nonfrictional decay of mesoscale eddies, J. Phys. Oceangr. 12, 244-255.
Smith, R. K: 1986, Evening glory wave-cloud lines in northwestern Australia, Austral. Meteor. Mag. 32, 27-33.
Smith, R. K: 1988, Traveling waves and bore in the lower atmosphere: The "morning glory" and related phenomena, Earth-Science Reviews 25,267-290. Review; weakly nonlocal internal gravity solitary waves.
Smith, R. K and Goodfield, J.: 1981, The 1979 morning glory expedition, Weather 36, 130-136.
Smith, R. K and Morton, B. R.: 1984, An observational study of northeasterly 'morning glory' wind surges, Austral. Meteor. Mag. 32, 155-175. Nonlocal internal gravity waves.
Smith, R. K and Noonan, J. A.: 1998, Generation of low-level mesoscale convergence lines over northeastern Australia, Mon. Weather Rev. In proof. This numerical study shows that mesoscale circulations force the Morning Glory gravity wave solitons as they propagate, compensating the solitary wave for strong radiative leakage to the upper atmosphere.
Smith, R. K, Coughlan, M. J. and Lopez, J.: 1986, Southerly nocturnal wind surges and bores in northeastern Australia, Mon. Weather Rev. 114, 1501-1518.
References 573
Smith, R. K., Crook, N. A. and Roff, G.: 1982, The Morning Glory: An extraordinary atmospheric undular bore, Quart. J. Royal Meteor. Soc. 108, 937-956. Nonlocal gravity soliton.
Smith, R. K., Reeder, M. J., Tapper, N. J. and Christie, D. R.: 1995, Central Australian cold fronts, Mon. Weather Rev. 123(1), 16-38. Role of central Australian fronts in triggering southerly Morning Glories (nonlocal undular bores).
Stenger, F.: 1993, Sinc Methods, Springer-Verlag, New York. Numerical, 500 pp.
Stengle, G.: 1977, Asymptotic estimates for the adiabatic invariance of a simple oscillator, SIAM J. Math. Anal. 8, 640-654.
Stengle, G.: 1985, Transcendental estimates for the adiabatic variation of linear Hamiltonian systems, SIAM J. Math. Anal. 16,932-940.
Stensrud, D. J.: 1987, Euclidean algorithm for finding common roots to polynomials, in H. N. Shirer (ed.), Nonlinear Hydrodynamic Modelling: A Mathematical Introduction, number 271 in Lecture Notes in Physics, SpringerVerlag, New York, part Appendix B, pp. 510-516.
Stern, M. E.: 1975, Minimal properties of planetary eddies, Journal of Marine Research 33, 1-13. Discovery of Rossby modons.
Sternin, B. Y. and Shatalov, V. E.: 1996, Borel-Laplace Transform and Asymptotic Theory: Introduction to Resurgent Analysis, CRC Press, New York.
Stieltjes, T. J.: 1886, Recherches sur quelques series semi-convergentes, Annales Scientifique de Ecole Normale Superieur 3, 201-258. Hyperasymptotic extensions to asymptotic series.
Sun, S. M.: 1991, Existence of a generalized solitary wave solution for water with positive Bond number less than 1/3, Journal of Mathematical Analysis with Applications 156, 471-504.
Sun, S. M.: 1998, On the oscillatory tails with arbitrary phase shift for solutions of the perturbed KdV equation, SIAM J. Appl. Math. Shows that the U-shaped curve of a versus the far field phase shift <I> is asymmetric, using higher order asymptoticsj this agrees with the numerical study of Champneys and Lord(1997).
Sun, S. M. and Shen, M. C.: 1993a, Exponentially small estimate for the amplitude of capillary ripples of a generalized solitary wave, Journal of Mathematical Analysis with Applications 172, 533-566.
574 WEAKLY NONLOCAL SOLITARY WAVES
Sun, S. M. and Shen, M. C.: 1993b, Exact theory of solitary waves in a stratified fluid with surface tension. Part H. Oscillatory case, J. Diff. Eq. 105(1), 117-166.
Sun, S. M. and Shen, M. C.: 1994a, Exponentially small asymptotics for internal solitary waves with oscillatory tails in a stratified fluid, Methods and Applications of Analysis 1(1),81-107.
Sun, S. M. and Shen, M. C.: 1994b, Exponentially small estimate for a generalized solitary wave solution to the perturbed K-dV equation, Nonlinear Analysis, Theory, Methods and Applications 23(4), 545-564. Proofs of the existence of the weakly nonlocal solitary wave and of the exponential smallness of its "wings" for the FKdV equation.
Sutyrin, G. G. and Dewar, W. K.: 1992, Almost symmetric solitary eddies in a two-layer ocean, J. Fluid Mech. 238, 633-. Rossby solitons.
Tan, B. and Boyd, J. P.: 1997, Dynamics of the Flierl-Petviashvili monopoles in a barotropic model with topographic forcing, Wave Motion 26, 239-252. Classical soli tons and Fourier pseudospectral methods.
Tanveer, S.: 1990, Analytic theory for the selection of Saffman-Taylor finger in the presence of thin-film effects, Proc. Roy. Soc. London A 428, 511-.
Tanveer, S.: 1991, Viscous displacement in a Hele-Shaw cell, in H. Segur, S. Tanveer and H. Levine (eds), Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 131-154.
Tepper, M.: 1950, A proposed mechanism of squall lines - the pressure jump line, Journal of Meteorology 7,21-29. First suggestion that undular bores (gravity wave solitons) might be important in the atmosphere.
Terril, R. M.: 1965, Laminar flow in a uniformly porous channel with large injection, Aeronautical Quarterly 16, 323-332.
Terrill, R. M.: 1973, On some exponentially small terms arising in flow through a porous pipe, Quart. J. Mech. Appl. Math. 26, 347-354.
Terrill, R. M. and Thomas, P. W.: 1969, Laminar flow in a uniformly porous pipe, Applied Scientific Research 21, 37-67.
Ticombe, M. S. and Ward, M. J.: 1997, Convective heat transfer past small cylindrical bodies, Stud. Appl. Math. 91, 81-105. A hybrid numericalasymptotic method is applied to compute the steady-state temperature around bodies of arbitrary shape. The log-and-powers series converges too slowly to be useful, but the hybrid algorithm has a broad range of accuracy while requiring only a small amount of computing.
References 575
Tin, S.-K: 1995, Transversality of double-pulse homoclinic orbits in the inviscid Lorenz-Krishnamurthy system, Physica D 83,383-408. Dynamical systems theory is applied to structures in the time coordinate which have the structure of weakly nonlocal solitary waves.
Ting, A. C., Chen, H. H. and Lee, Y. C.: 1987, Exact solutions of a nonlinear boundary value problem: the vortices of the two dimensional sinh-Poisson equation, Physica D 26, 36-66.
Tovbis, A.: 1994, On exponentially small terms of solutions to nonlinear ordinary differential equations, Methods and Applications of Analysis 1(1), 57-74.
Tovbis, A., Tsuchiya, M. and Jaffe, C.: 1998, Exponential asymptotic expansions and approximations of the unstable and stable manifolds of the Henon map, Physica D. submitted.
Trefethen, L. N. and Trummer, M. R.: 1987, An instability phenomena in spectral methods, SIAM J. Numer. Anal. Exponential sensitivity of eigenvalues.
Tribbia, J. J.: 1984, Modons in spherical geometry, Geophys. Astrophys. Fluid Dyn. 30, 131-168. Nonlocal Rossby waves.
Tu, Y.: 1991, Saffman-Taylor problem in a sector geometry, in H. Segur, S. Tanveer and H. Levine (eds), Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 175-186.
Tung, K K, Chan, T. F. and Kubota, T.: 1982, Large-amplitude internal waves of permanent form, Stud. Appl. Math. 66, 1-44. Analytical theory combined with direct numerical solutions of two-dimensional nonlinear eigenvalue problem.
Tung, K K, Ko, R. S. and Chang, J. J.: 1981, Weakly nonlinear internal waves in shear, Stud. Appl. Math. 65, 189-221. Analytical theory combined with direct numerical solutions of two-dimensional nonlinear eigenvalue problem.
Turkington, B. and Whitaker, N.: 1996, Statistical equilibrium computations of coherent structures in turbulent shear layers, SIAM J. Sci. Comput. 17(6), 1414-1433. Numerical calculations, using a variational iteration, of classical coherent structures.
Turkington, B., Eydeland, A. and Wang, S.: 1991, A computational method for solitary internal waves in a continuously stratified fluid, Stud. Appl.
576 WEAKLY NONLOCAL SOLITARY WAVES
Math. 85, 93-127. Non-Newtonian iteration for computing solitary waves, based on optimization theory; fast and effective.
Turner, R. E. L.: 1991, Waves with oscillatory tails, Nonlinear Analysis: a Tribute in Hone of Giovani Prodi, Scuola Normale Superiore, Pisa, pp. 335-348.
Ursell, F.: 1990, Integrals with a large parameter. A strong form of Watson's lemma, in R. W. Ogden and G. Eason (eds), Elasticity: Mathematical Methods and Applications, Ellis Horwood Limited, pp. 391-395.
Vallis, G. K.: 1985, On the spectral integration of the quasi-geostrophic equations for doubly-periodic and channel flow, J. Atmos. Sci. 42, 95-99. Timeintegration with Fourier basis.
Vallis, G. K., Carnevale, G. F. and Shepherd, T. G.: 1990, A natural method for finding stable states of Hamiltonian systems, in H. K. Moffatt (ed.), Proceedings of the IUTAM Conference on Topological Fluid Dynamics, Cambridge University Press, New York, pp. 429-439. Non-Newtonian numerical method for computing solitons.
Vallis, G. K., Carnevale, G. F. and Young, W. R.: 1989, Extremal energy properties and construction of stable solutions of the Euler equations, J. Fluid Mech. 207, 133-152. Enstrophy-preserving Hamiltonian dissipation which monotonically increases or decreases energy to find nonlinear steady or steadily-translating states, here applied to two-dimensional flow with rotation. Slow, but generates intriguing, stable structures.
Van Dyke, M.: 1975, Perturbation Methods in Fluid Mechanics, 2d edn, Parabolic Press, Stanford, California.
Van Dyke, M.: 1994, Nineteenth-century roots of the boundary-layer idea, SIAM Rev. 36, 415-424.
Vanden-Broeck, J.-M.: 1983, Fingers in Hele-Shaw cell with surface tension, Phys. Fluids 26, 2033-2034. Exponential smallness in a non-soliton phenomenon.
Vanden-Broeck, J.-M.: 1984, Rising bubbles in a two-dimensional tube with surface tension, Phys. Fluids 27, 2604-2607. Exponential smallness in a non-soliton phenomenon.
Vanden-Broeck, J.-M.: 1986, A free streamline model for a rising bubble, Phys. Fluids 29, 2798-2801.
References 577
Vanden-Broeck, J.-M.: 1988, A free streamline model for a rising bubble, Phys. Fluids 31, 974-977. Exponential smallness in a non-soliton phenomenon.
Vanden-Broeck, J.-M.: 1991a, Elevation solitary waves with surface tension, Phys. Fluids A 3, 2659-2663. Numerical solutions for weakly nonlocal water waves.
Vanden-Broeck, J.-M.: 1991b, Gravity-capillary free surface flows, in H. Segur, S. Tanveer and H. Levine (eds), Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 275-29l.
Vanden-Broeck, J.-M.: 1992, Rising bubble in a two-dimensional tube: asymptotic behavior for small values of the surface tension, Phys. Fluids 35, 2332-2334. Exponential smallness in a non-soliton phenomenon.
Vanden-Broeck, J.-M. and Dias, F.: 1992, Gravity-capillary solitary waves in water of infinite depth and related free surface flows, J. Fluid Mech. 240, 549-557.
Vanden-Broeck, J.-M. and Turner, R. E. L.: 1992, Long periodic internal waves, Phys. Fluids A 4, 1929--1935.
Verkley, W. T. M.: 1984, The construction of barotropic modons on a sphere, J. Atmos. Sci. 41, 2492-2504.
Verkley, W. T. M.: 1987, Stationary barotropic modons in westerly background flows, Journal of the Atmospheric Sciences 44, 2383-2398. Nonlocal modons ..
Verkley, W. T. M.: 1990, Modons with uniform absolute vorticity, J. Atmos. Sci. 47, 727-745.
Verkley, W. T. M.: 1993, A numerical method to find form-preserving free solutions of the barotropic vorticity equation on a sphere, J. Atmos. Sci. 50, 1488-1503. Non-Newtonian iteration to compute solitary waves and steady nonlinear flows.
Wai, P. K. A., Chen, H. H. and Lee, Y. C.: 19, Radiation by "solitons" at the zero group-dispersion wavelength of single-mode optical fibers, Phys. Rev. A 41, 426-439. Nonlocal envelope solitons of the TNLS Eq. Their (2.1) contains a typo and should be q(2) = -(39/2)A2q(O) + 21jq(O) j2q(O).
Wai, P. K. A., Menyuk, C. R., Lee, Y. C. and Chen, H. H.: 1986, Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers, Optics Lett. 11, 464-466.
578 WEAKLY NONLOCAL SOLITARY WAVES
Ward, M. J.: 1994, Metastable patterns, layer collapses, and coarsening for a one-dimensional Ginzburg-Landau equation, Stud. Appl. Math. 91, 51-93. Log-and-power series.
Ward, M. J. and Reyna, L. G.: 1995, Internal layers, small eigenvalues, and the sensitivity of metastable motion, SIAM J. Appl. Math. 55, 426-446. Allen-Cahn and Burgers equations.
Ward, M. J., Henshaw, W. D. and Keller, J. B.: 1993, Summing logarithmic expansions for singularly perturbed eigenvalue problems, SIAM J. Appl. Math. 53, 799-828.
Warn, T.: 1997, Nonlinear balance and quasi-geostrophic sets., AtmosphereOcean 35, 135-145. Written circa 1982, this "grey literature" classic influenced much later work on the slow manifold. Warn seems to have been the first to suspect and offer good theoretical reasons why the multiple scales initialization for numerical weather prediction is divergent. This article was not published at the time of writing because the divergence of a numerical procedure does not conclusively prove the nonexistence of its target - in this case, the slow manifold.
Wasserstrom, E.: 1973, Numerical solutions by the continuation method, SIAM Rev. 15, 89-119.
Wazwaz, A. M. and Hanson, F. B.: 1991, Moments of extinction for a singular perturbation problem in the diffusion process, SIAM J. Appl. Math. 51(1), 233-265. Exponential asymptotics.
Weideman, J. A. C.: 1992, The eigenvalues of Hermite and rational spectral differentiation matrices, Numerische Math. 61, 409-431.
Weideman, J. A. C. and Cloot, A.: 1990, Spectral methods and mappings for evolution equations on the infinite line, Comput. Meth. Appl. Mech. Engr. 80, 467-481. Numerical.
Weideman, J. A. C. and James, R. L.: 1992, Pseudospectral methods for the Benjamin-Ono equation, Advances in Computer Methods for Partial Differential Equations VII pp. 371-377. Both Fourier series and rational functions on the infinite interval are used for the spatial discretization.
Weideman, P. D.: 1978a, Internal solitary waves in a linearly stratified fluid, Tellus 30, 177-184. Classical solitons.
Weideman, P. D.: 1978b, Corrigendum to the paper Internal solitary waves in a linearly stratified fluid, Tellus 31, 465-467.
References 579
Weinstein, M. 1. and Keller, J. B.: 1985, Hill's equation with a large potential, SIAM J. Appl. Math. 45, 200-214.
Weinstein, M. 1. and Keller, J. B.: 1987, Asymptotic behavior of stability regions for Hill's equation, SIAM J. Appl. Math. 47, 941-958.
Weniger, E. J.: 1989, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Computer Physics Reports 10, 189-371.
Weniger, E. J.: 1991, On the derivation of iterated sequence transformations for the acceleration of convergence and the summation of divergent series, Comput. Phys. Commun. 64, 19-45.
Williams, G. P.: 1996, Jovian dynamics. Part 1. Vortex stability, structure and genesis, J. Atmos. Sci. 53(18), 2685-2734. Fig. 1 is a good illustration of radiative decay of baroclinic equatorial Rossby solitons, theoretically predicted by Marshall and Boyd(1987), through leakage into the barotropic mode. The solitary wave remains coherent for a very long time in spite of the leakage.
Williams, G. P. and Wilson, R J.: 1988, The stability and genesis of Rossby vortices, J. Atmos. Sci. 45, 207-249. Show that n = 3 and higher mode Rossby solitons radiatively decay, and appear to be examples of ''weakly nonlocal" solitons.
Wimp, J.: 1961, Polynomial approximations to integral transforms, Math. Comp. 15, 174-178. Chebyshev series.
Wimp, J.: 1962, Polynomial expansions of Bessel functions and some associated functions, Math. Comp. 16, 446-458.
Wimp, J.: 1967, The asymptotic representation of a class of G-functions for large parameter, Math. Comp. 21, 639-646.
Wimp, J.: 1981, Sequence Transformations and Their Applications, Academic Press, New York.
Wong, R: 1989, Asymptotic Approximation of Integrals, Academic Press, New York.
Wong, R (ed.): 1990, Asymptotic and Computational Analysis, Marcel Dekker, New York. Many papers on exponential asymptotics.
Wood, A. D.: 1991, Exponential asymptotics and spectral theory for curved optical waveguides, in H. Segur, S. Tanveer and H. Levine (eds), Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 317-326.
580 WEAKLY NONLOCAL SOLITARY WAVES
Wood, A. D. and Paris, R B.: 1990, On eigenvalues with exponentially small imaginary part, in R Wong (ed.), Asymptotic and Computational Analysis, Marcel Dekker, New York, pp. 741-749.
Yanase, S. and Nishiyama, K.: 1988, On the bifurcation of laminar flows through a curved rectangular tube, J. Phys. Soc. Japan 57, 3790-3795. Compute two-dimensional steady flow via a simple, non-Newtonian iteration that requires inverting only the Chebyshev pseudospectral matrix for the Laplace operator.
Yang, J.: 1996a, Coherent ~.structures in weakly birefringent nonlinear optical fibers, Stud. Appl. Math. 97, 127-148. Explicit computation of the far field oscillations for nonlocal solitary waves in a system of two coupled Nonlinear Schrodinger equations.
Yang, J.: 1996b, Vector solitons and their internal oscillations in birefringent nonlinear optical fibers, Stud. Appl. Math. 98, 61-97. Explicit computation of the far field oscillations ("internal oscillations") for nonlocal solitary waves in a system of two coupled Nonlinear Schrodinger equations.
Yang, T.-S.: 1998, On traveling-wave solutions of the Kuramoto-Sivashinsky equation, Physica D. Shocks with oscillations, exponentially small in liE, which grow slowly in space, and thus are (very!) nonlocal. Applies the Akylas-Yang beyond-aIl-orders perturbation method in wavenumber space to compute the far field for oscillatory shocks. These are then matched to the nonlocal regular shocks to create solitary waves (that asymptote to the same constant as x -t ±oo); these are confirmed by numerical solutions.
Yang, T.-S. and Akylas, T. R: 1995, Radiating solitary waves of a model evolution equation in fluids of finite depth, Physica D 82, 418-425. Solve the Intermediate-Long Wave (ILW) equation for water waves with an extra third derivative term, which makes the solitary waves weakly nonlocal. The Yang-Akylas matched asymptotics in wavenumber is used to calculate the exponentially small amplitude of the far field oscillations.
Yang, T.-S. and Akylas, T. R: 1996a, Weakly nonlocal gravity-capillary solitary waves, Phys. Fluids 8, 1506-1514.
Yang, T.-S. and Akylas, T. R: 1996b, Finite-amplitude effects on steady leewave patterns in subcritical stratified flow over topography, J. Fluid Mech. 308,147-170.
Yang, T.-S. and Akylas, T. R: 1997, On asymmetric gravity-capillary solitary waves, J. Fluid Mech. 330, 215-232. Asymptotic analysis of classical
solitons of the FKdV equation; demonstrates the coalescence of classical bions.
Yih, C.-S.: 1968, Fluid Mechanics, West River Press, Ann Arbor.
Zangwill, W. 1. and Garcia, C. B.: 1981, Pathways to Solutions, Fixed Points, and Equilibria, Computational Mathematics, Prentice-Hall, Englewood Cliffs, New Jersey. Very readable introduction to continuation methods for solving systems of nonlinear equations. Title page lists Zangwill first but spine lists Garcia first.
Zaslavsky, G. M., Sagdeev, R. Z., Usikov, D. A. and Chernikov, A. A.: 1991, Weak Chaos and Quasi-Regular Patterns, Cambridge University Press, New York.
Zhu, Y. and Dai, S.: 1991, On head-on collision between two GKdV solitary waves in a stratified fluid, Acta Mechanica Sinica 7(4), 300-308. Their GKdV equation is actually the Fifth-Order Korteweg-deVries equation.
Index
cjy4 breathers Bibliography Tables eigenfunctions of Newton-Kantorovich Classical FKdV, 247
Eq., 316-318 Classical Gravity Waves, 280 far field analysis, 314-318 Classical Modons, 408 multiple scales series, 310-314 Classical Rossby Waves, 282 numerical calculations, 318-321 Existence and Nonexistence Proofs, phase factor <Pmin, 321 134 phase factor <Pres , 320 Exponentially Small Quantum radiative decay, 321-323 Phenomena, 457 table of multiple scales series Gulf Stream Rings & Related
coefficients, 313 Vortices, 424
Above-the-barrier wave reflection (scattering), 83-88
AEW (Ageostrophic Equatorial Wave) Eq.
application to Rossby waves, 294-301
definition, 285 Assertions
Exponential Smallness, 21 Phase-Speed Matching, 18 Super asymptotic Error Equals
0:,60 Asymptotic
(Definition), 48
BDO (Benjamin-Davis-Ono) Eq. gravity waves & Morning Glory,
406 soliton identities, 496 spatially-periodic exact solution,
499 Bezout's Theorem (number of roots
of polynomial system, 221
582
LK Quintet (Recent Studies), 383
Morning Glory, 404 Non-Soliton Exponential Small
ness, 456 Nonlocal Breathers, 309 Nonlocal Capillary-Gravity Wa-
ter Waves, 246 Nonlocal Envelope Solitons, 327 Nonlocal FKdV, 245 Nonlocal Internal Gravity Waves,
280 Nonlocal Modons, 412 Nonlocal Rossby Waves, 281 Nonlocal Solitary Waves, 8 Radiative Decay, 435 Resurgence and Related Stud-
ies, 49 RMKdV & RMKP Eqs., 397 Slow Manifold, 380 Time-Marching with a Fourier
Pseudospectral Method, 501
Index
cnoidal matching, 235-238 Complex plane-matched asymptotics,
24-25,80-105 continuation, 182-195
complex-plane, 194-195 continuing around fold points,
191 initialization by "persistence",
184 initialization by polynomial ex
trapolation, 188
583
Tunnelling [Root-Finding] Algorithm, 203
dendritic crystal growth, 455-459 direct root-finding methods for spe
ciallow order systems, 219-221
Dubreil-Jacotin-Long Eq. application to internal gravity
waves, 283 definition, 283
shooting bifurcation (branch-crossing) eigenfunctions of N ewton-Kantorovich point, 192 Eq.
Coupled-KdV cjJ4 breathers, 316-318 seeK dV-Coupled Two-Mode Sys- FKdV Eq., 257-262
tern, 1 TNLS Eq., 341-343
Davidenko equation, 176-177, 185-188
Definitions Asymptotic, 48 Breather, 306 Classical Solitary Wave, 11 Envelope Solitary Wave, 325 Far field, 14 Far Field Phase <1>, 14 fold point, 189 Geometric convergence, 150 Hyperasymptotic, 51 limit point, 189 Micropteron, 11 Monopole Vortex, 421 Nanopteroidal Wave, 12 Nanopteron, 11 Nonlocal Solitary Wave, 11 Parity, 159 Radiation Coefficient, 14 Radiatively Decaying Soliton,
11 residual function, 142 Subgeometric convergence, 151 Superasymptotic, 50
elliptic functions, 489-493 envelope solitary wave
definition, 325 equatorial waves
Kelvin wave instability with exponentially small growth rate, 472
leakage-to-barotropic mode, 427 nonlocal Rossby solitons, 426-
429 Euler sum-acceleration, 56 Existence proofs, 135-136
Bibliography Table, 134 Exponential Smallness of the Radi
ation Coefficient Assertion, 21 Theorem, 38
Eydeland-Turkington iteration for Rossby solitons, 215
Fallacies Periodic-Irrelevant-to-Solitons,
5 Solitons-Are-Rare, 5
Far Field FKdV Eq., 15-18
584
Far Field Phase, 16-18 (Definition), 14 relationship to spatial period,
228-229 FKdV (Fifth-Order Korteweg-de Vries)
Eq., 15, 243-278 Bibliography Table:Classical, 247 Bibliography Table:Nonlocal, 245 cnoidal wave
regimes, 122-126 relationship with polycnoidal
waves, 127 Stokes series, 111-112
complex plane-matched asymptotics, 264-265
derivation for capillary-gravity water waves, 274-277
eigenfunctions, 257-260 Generic derivation, 246-249 minimum radiation coefficient,
260 multiple scales perturbation the
ory,250-255" numerical solutions, 266 perturbation series coefficients
(Table), 250 Pseudo code for perturbation se
ries, 251 radiative decay, 268-271 Rescaling to coefficients of unit
magnitude, 249 resonance & chaos, 260-262 symmetry, 262-264
fold point, 189-192 (Definition), 189 continuation around by inter
change of parameters, 191 continuation around by pseu
doarclength continuation, 191
Galerkin's method
INDEX
(Definition), 143 merits versus pseudospectral, 144
genetic algorithms, 207 gravity waves
laboratory experiments for nonlocal solitons, 302
model equations, 283 physics background, 279
Gulf Stream Rings oceanic background, 422
Gulf Stream rings, 7-9
HNOR (Hyperasymptotic-with-(N + 1 )st-Order-Residual) Approximati(Definition)
(Definition), 70 HOTR (Hyperasymptotic-with-One
Term-Residual) Approximation
(Definition), 71 extrapolation, 73
Hyperasymptotic Perturbation The-ory
(Definition), 51 Nonlocal Solitons, 57-79 Stieltjes function, 54-57
implicitly-implicit wave equations Fourier pseudospectral algorithm
for, 506 initialization procedure
connections with solitons, 385 weather prediction, 367
instability with exponentially small growth rate, 473
inverse scattering method, 5-7
KdV (Korteweg-deVries) Eq. cnoidal wave formulas, 496 derivation by eigenfunction ex
pansion,285-288 derivation by multiple scales,
295-296
Index
soliton identities, 494 KdV with Mixed Cubic and Quadratic
Nonlinearity model for QH Eq., 292-293 soliton identities, 495
KdV-Coupled (Two-Mode) System definition, 290 equatorial Rossby waves, 296-
297 gravity waves, 289-292
KdVB (Korteweg-deVries-Burgers) Eq. shocks, 487
KG (Klein-Gordon) Eq. Bibliography Table, 327 Fourier pseudospectral method
in group and carrier coordinates, 355-357
KG Lorentz Invariance Theo-rem, 354
KG Scaling Theorem, 354 numerical solutions , 360-363 resonant spatial period, 359 spatial period for minimum a,
359
Levenberg-Marquadt root-finding method, 202
limit point (Definition), 189
LK Quintet Bibliography Tables of Recent
Studies, 383 chaos around separatrices, 384 dynamical systems theory, 383-
384 One-Way-Coupled Approximate
Model, 376, 383 physics background, 375-379 slow manifold defined, 377
log-contour plot root-finding method, 219
Long's Eq.
585
seeDubreil-Jacotin-Long Eq., 1
Matched asymptotics seeComplex plane-matched asymp
totics, 1 Maxwell-Bloch Eqs., 348 Micropteron
(Definition), 11 micropterons
as generic consequence of long wave resonance, 387
MKdV (Modified Korteweg-de Vries) Eq.
cnoidal and dnoidal exact solutions, 498
equatorial antisymmetric Rossby waves, 297-301
soliton identities, 495 modons
magnetic in shear, 301-302 monopole vortex
definition, 421 Gulf Stream Rings, 422 in Rossby waves, 422-429
Morning Glory gravity solitons Benjamin-Davis-Ono (BDO) Eq.,
406-407 Bibliography Table, 404 meteorological background, 399-
402 vertical trapping, 403
Multiple Scales Perturbation Theory
<jJ4 breather example, 310-314 conversion of sech/tanh pow
ers to ordinary polynomials, 486
divergence, 34, 40 FKdV example, 22-24, 250-255 TNLS Eq. example, 337-341
Nanopteroidal Wave
586
(Definition), 12 Nanopteron
(Definition), 11 as a short wave resonance, 388 differences in resonant wavenum-
ber from micropteron, 388 Newton flow, 176 Newton's iteration, 174-175
branch-switching, 194 Exponential Decay of Newton
Flow Theorem, 199 geometry of phase space, 197 hyperasymptotic perturbation
theory, 62-64 initialization by abitrary guess,
180 initialization by Davidenko pre
dictor, 175 initialization by intrinsically non
linear approximation, 178 initialization by low order spec
tral method, 181 initialization by polynomial ex
trapolation, 175 list of flaws, 196 optimization and minimization
of residual norm, 197 quasi-Newton methods, 208-212 underrelaxation, 198-202
Newton's method Newton flow equation, 176
NLS (N onlinear Schroedinger) Eq. connection with TNLS Eq., 328 soliton identities, 495
Non-analyticity of the Radiation Coefficient
(Theorem), 39 Non-Newtonian root-finding itera
tions, 212-218 Non-soliton exponential smallness
Bibliography (non-quantum), 456
INDEX
dendrites on solid-liquid interface, 455
equatorial Kelvin wave instability, 471
errors in trapezoidal rule inte-gration, 470
flow in porous pipe, 476-478 linear eigenproblems, 471-475 Quantum Bibliography, 457 quantum Stark effect, 461 Saffman-Taylor problem, 459 steepest descent asymptotics for
integrals, 464 ultra-slow diffusion of phase-transition
fronts, 463 viscous fingering in a thin fluid
layer, 459 Nonexistence proofs, 133-135
Bibliography Table, 134 Nonlocal Modons
applied to atmospheric blocking, 412
basic modon theory, 407-412 bibliography of classical mod-
ons, 408 Bibliography Table, 412 oceanic background, 407 Swaters-nolocal-to-the-east type,
417-421 Tribbia-Verkley-Boyd type, 412-
417 numerical analysis, exponential small
ness in, 468-471
Optimal Truncation Rule of Thumb, 48 Rule of Thumb: Fourier Trans
form Heuristic, 75 Rule of Thumb: One-Term Resid
ual Heuristic, 76
Pade approximants, 44
Index
parity (Definition), 159
pendulum, simple, 368-373 Petviashvili root-finding iteration, 216 Phase-Speed Matching Rule
Assertion, 18 FKdV illustration, 19-20
Poljak flow root-finding method, 205 Polycnoidal waves, 127-130 porous pipe flow, 476-478 pseudodata, 367 pseudospectral method
(Definition), 142 accuracy, 150-155 Bibliography of Fourier Initial-
value Algorithms, 501 choice of basis set, 145-149 cnoidal matching, 235-238 continuous symmetries, 161-164 dangers in library Fast Fourier
Transform software, 505 embedding in finite difference
code, 159 envelope of the spectral coeffi
cients, 152 Fast Fourier Transform for ini
tial value calculations, 502 Fourier basis for nonlocal soli
tary waves, 225-229 geometric convergence (Defini
tion), 150 implicitly-implicit wave equa-
tions (RLW, quasi-geostrophic), 506
infinite interval basis sets (Ta-ble), 149
initial value problems, 500-507 KdVB shock example, 167 MATLAB code for KdV initial
value problem, 505 merits versus Galerkin's, 144 parity, 159-161
587
Quasi-Sinusoidal Rule-of-Thumb, 154
radiation function basis, 232-234
Rational Chebyshev functions (Subroutine), 147
shocks and kinks, 167 solving initial-value problems when
a spatial differential operator acts on the time derivatives
tor acts on the time derivatives, 506
subgeometric convergence (Definition), 151
pseudospectral methods multidimensional basis sets, 149
Purification-by-Dispersion (to compute solitons), 213
QH (Quadratic Helmholtz) Eq. defintion, 284
QHLS (Quadratic Helmholtz with Linear Stratification) Eq.
definition, 284 eigenseries analysis, 285-289
quantum phenomena above-the-barrier scattering (ex
ponentially weak reflection, 83-89
Bibliography of exponential smallness, 457
Stark effect (exponentially slow tunnelling decay), 461
quenching (enstrophy-preserving Hamiltonian damping, 217
radiation basis function pseudocode to compute for cnoidal
matching algorithm, 236 Radiation Coefficient Cl!
(Definition), 14
588
exponential smallness, 38 forced, constant coefficient ODE,
64-66 non-analyticity of, 39 sensitivity to perturbations, 44 table for linear constant coeffi-
cient differential eqs., 65 radiation coefficient 0:
sensitivity, 238-239 radiative decay of nonlocal solitons
</14 breather, 268, 321, 443 Bibliography, 435 forced, linear PDE example, 440 micropterons, 451 ODEs in time for decay, 447-
453 one-sided vs. two-sided, 434,
440,454 perturbation theory, 435-440
KdV double soliton example, 444-446
role of dispersion, 443 spike in Fourier spatial trans
form at k = kf' 443 TNLS envelope soliton, 349
rational Chebyshev functions, 147-149, 229-232
KdVB shock example, 167 subroutine to compute, 147 with radiation basis function,
232 residual function (Definition), 142 result ants for solving low order poly
nomial systems, 220 RMKdV (Rotation-Modified Korteweg
de Vries) Eq. as simplest example of micropteron,
389 Background, 389 Bibliography Table, 397 far field analysis, 390
INDEX
Matched asymptotics calculation of the radiation coefficient, 392
numerical solutions, 394 Rescaling Theorem, 390 spatial period for resonance (0: ~
00),394 spatial period of minimum 0:,
394 Zero Mean Theorem, 391
RMKP (Rotation-Modified KadomtsevPetviashvili) Eq.
applied to gravity waves in fluids, 396-399
background, 396-398 Bibliography Table, 397
Rossby waves Gulf Stream Ring Bibliography
Table, 424 Gulf Stream Rings, 422-426 leakage through low-latitude ra-
diation, 426 leakage-to-barotropic mode, 427 model equations, 283 numerical observations of non-
linear equatorial solitons, 303
physics background, 280 Rule of Thumb
Optimal truncation, 48 Optimal truncation: Fourier Trans
form Heuristic, 75 Optimal truncation: One-Term
Residual Heuristic, 76 Quasi-Sinusoidal (pseudospec-
tral accuracy), 154
Saffman-Taylor viscous fingering, 460 separatrix splitting, 369-373
connection with nonlocal solitons, 372
simulated annealing, 206
Index
slow manifold (possible) exponential smallness
of gravity waves, 385 connection with solitons, 385 LK Quintet, 375 multiple scales in time, 379-382 physics background, 373-375 Table of Eight Definitions of Slow
Manifold, 380 Solitary Wave
Classical (Definition), 11 Weakly Nonlocal (Definition),
11 Soliton
Radiatively Decaying (Definition), 11
"Soliton" -for-Nonintegrable-Solitary Wave Controversy, 248
soliton and sine wave regimes, overlap of, 106-108, 114-116
Soliton/Solitary Wave Terminological Controversy, 248
steepest descent method for integrals, 465-468
Stokes expansion accuracy, 114 application to cnoidal match
ing, 235 FKdVexample, 108-112 overlap with approximation-by
soliton, 115 radius of convergence, 112-114 resonances (small denominators),
112-114 Stokes' phenomenon in asymptotics,
smoothing by hyperasymptotics, 465
Superasymptotic Error Equals a Assertion, 60
Symmetry of nonlocal solitary waves theory, 136-138
589
Tables Analytical Initialization, 177 Basis Sets for the Infinite In
terval, 149 Cost-Reducing Strategies (New
ton's Iteration), 208 Direct Methods for Special Non
linear Systems, 219 Domain-of-Convergence-Increasing
Strategies, 203 Errors in Approximation of a
by Radiation Basis Function Method
Method,234 Errors in Numerical Approxi
mation of a, 234 FKdV multiple scales coefficients,
250 FKdV Pseudocode for pertur
bation series, 251 Fourier pseudospectral MATLAB
code for KdV Eq., 505 Matlab Subroutine to Compute
Rational Chebyshev Basis Functions, 147
Non-Newtonian Iterations, 212 Pseudocode to Compute Radi
ation Basis Function, 236 Radiation Coefficient a for Forced,
Constant Coefficient ODE ODE,65
Tunnelling Strategies, 204 Theorems
Divergence of Multiple Scales Series, 40
Error in Asymptotic Series, 42 Exponential Decay of Newton
Flow, 199 Exponential Smallness of the Ra
diation Coefficient, 39 Imaginary Part of Eigenvalue,
474
590
Klein-Gordon Lorentz Invariance, 354
Klein-Gordon Scaling, 354 Non-analyticity of the Radia-
tion Coefficient, 39 Parity, 160 Relationship Between Multiple
Scales and Fourier Integrals, 41
RMKdV Rescaling, 390 RMKdV Zero Means, 391 TNLS Carrier Wavenumber Shift,
335 TNLS Scaling, 333
INDEX
tunnelling (away from a minimum of the residual norm), 203
TNLS (Third-Order Nonlinear Schroedinger) Eq.
"all orders" eigenrelation, 337 Bibliography Table, 327 Carrier Wavenumber Shift The-
orem, 335 connection with NLS Eq., 328 differences from FKdV nanopterons,
332 eigenfunctions of the N ewton
Kantorovich Eq., 341-343 multiple scales perturbation the
ory, 337-339 nanopteroidal waves, 339-341 nonlocal bions (bound state of
pairs of solitons), 348-349 numerical solutions, 344-347 physics background, 328-331 radiative decay, 349-352 resonant spatial period, 343 Scaling Theorem, 333 similarities to FKdV nanopterons,
331 spatial period of minimum 0:,
343 summary, 331
"trust region" root-finding method, 202
Other Mathematics and Its Applications titles of interest:
I.H. Dimovski: Convolutional Calculus. 1990,208 pp. ISBN 0-7923-0623-6
Y.M. Svirezhev and V.P. Pasekov: Fundamentals of Mathematical Evolutionary Genetics. 1990,384 pp. ISBN 90-277-2772-4
S. Levendorskii: Asymptotic Distribution of Eigenvalues of Differential Operators. 1991,297 pp. ISBN 0-7923-0539-6
V.G. Makhankov: Soliton Phenomenology. 1990,461 pp. ISBN 90-277-2830-5
I. Cioranescu: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. 1990,274 pp. ISBN 0-7923-0910-3
B.I. Sendov: Hausdorff Approximation. 1990,384 pp. ISBN 0-7923-090 1-4
A.B. Venkov: Spectral Theory of Automorphic Functions and Its Applications. 1991,280 pp. ISBN 0-7923-0487-X
V.I. Arnold: Singularities of Ca us tics and Wave Fronts. 1990,274 pp. ISBN 0-7923-1038-1
A.A. Pankov: Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations. 1990,232 pp. ISBN 0-7923-0585-X
A.S. Davydov: Solitons in Molecular Systems. Second Edition. 1991,428 pp. ISBN 0-7923-1029-2
B.M. Levitan and I.S. Sargsjan: Sturm-Liouville and Dirac Operators. 1991, 362 pp. ISBN 0-7923-0992-8
V.I. Gorbachuk and M.L. Gorbachuk: Boundary Value Problems for Operator Differential Equations. 1991,376 pp. ISBN 0-7923-0381-4
Y.S. Samoilenko: Spectral Theory of Families of Self-Ad joint Operators. 1991, 309 pp. ISBN 0-7923-0703-8
B.I. Golubov A.V. Efimov and V.A. Scvortsov: Walsh Series and Transforms. 1991,382 pp. ISBN 0-7923-1100-0
V. Laksmikantham, V.M. Matrosov and S. Sivasundaram: Vector Lyapunov Functions and Stability Analysis of Non linear Systems. 1991,250 pp.
ISBN 0-7923-1152-3
F.A. Berezin and M.A. Shubin: The Schrodinger Equation. 1991,556 pp. ISBN 0-7923-1218-X
D.S. Mitrinovic, J.E. Pecaric and A.M. Fink: Inequalities Involving Functions and their Integrals and Derivatives. 1991,588 pp. ISBN 0-7923-1330-5
Julii A. Dubinskii: Analytic Pseudo-Differential Operators and their Applications. 1991,252 pp. ISBN 0-7923-1296-1
V.I. Fabrikant: Mixed Boundary Value Problems in Potential Theory and their Applications. 1991,452 pp. ISBN 0-7923-1157-4
Other Mathematics and Its Applications titles of interest:
A.M. Samoilenko: Elements of the Mathematical Theory of Multi-Frequency Oscillations. 1991,314 pp. ISBN 0-7923-1438-7
Yu.L. Dalecky and S.V. Fomin: Measures and Differential Equations in InfiniteDimensional Space. 1991,338 pp. ISBN 0-7923-1517-0
W. Mlak: Hilbert Space and Operator Theory. 1991,296 pp. ISBN 0-7923-1042-X
N.Ja. Vilenkin and A.U. Klimyk: Representation of Lie Groups and Special Functions. Volume 1: Simplest Lie Groups, Special Functions, and Integral Transforms. 1991,608 pp. ISBN 0-7923-1466-2
N.Ja. Vilenkin and A.U. Klimyk: Representation of Lie Groups and Special Functions. Volume 2: Class I Representations, Special Functions, and Integral Transforms. 1992, 630 pp. ISBN 0-7923-1492-1
N.Ja. Vilenkin and A.U. Klimyk: Representation of Lie Groups and Special Functions. Volume 3: Classical and Quantum Groups and Special Functions. 1992, 650 pp. ISBN 0-7923-1493-X
(Set ISBN for Vols. 1,2 and 3: 0-7923-1494-8)
K. Gopalsamy: Stability and Oscillations in Delay Differential Equations of Population Dynamics. 1992,502 pp. ISBN 0-7923-1594-4
N.M. Korobov: Exponential Sums and their Applications. 1992,210 pp. ISBN 0-7923-1647-9
Chuang-Gan Hu and Chung-Chun Yang: Vector-Valued Functions and their Applications. 1991, 172 pp. ISBN 0-7923-1605-3
Z. Szmydt and B. Ziemian: The Mellin Transformation and Fuchsian Type Partial Differential Equations. 1992, 224 pp. ISBN 0-7923-1683-5
L.I. Ronkin: Functions of Completely Regular Growth. 1992, 394 pp. ISBN 0-7923-1677-0
R. Delanghe, F. Sommen and V. Soucek: Clifford Algebra and Spinor-valued Functions. A Function Theory of the Dirac Operator. 1992, 486 pp.
ISBN 0-7923-0229-X
A. Tempelman: Ergodic Theorems for Group Actions. 1992, 400 pp. ISBN 0-7923-1717-3
D. Bainov and P. Simenov: Integral Inequalities and Applications. 1992,426 pp. ISBN 0-7923-1714-9
I. Imai: Applied Hyperfunction Theory. 1992, 460 pp. ISBN 0-7923-1507-3
Yu.1. Neimark and P.S. Landa: Stochastic and Chaotic Oscillations. 1992,502 pp. ISBN 0-7923-1530-8
H.M. Srivastava and R.G. Buschman: Theory and Applications of Convolution Integral Equations. 1992,240 pp. ISBN 0-7923-1891-9
Other Mathematics and Its Applications titles of interest:
A. van der Burgh and J. Simonis (eds.): Topics in Engineering Mathematics. 1992, 266 pp. ISBN 0-7923-2005-3
F. Neuman: Global Properties of Linear Ordinary Differential Equations. 1992, 320 pp. ISBN 0-7923-1269-4
A. Dvurecenskij: Gleason's Theorem and its Applications. 1992, 334 pp. ISBN 0-7923-1990-7
D.S. Mitrinovic, J.E. Pecaric and A.M. Fink: Classical and New Inequalities in Analysis. 1992,740 pp. ISBN 0-7923-2064-6
H.M. Hapaev: Averaging in Stability Theory. 1992,280 pp. ISBN 0-7923-1581-2
S. Gindinkin and L.R. Volevich: The Method of Newton's Polyhedron in the Theory ofPDE's. 1992,276 pp. ISBN 0-7923-2037-9
Yu.A. Mitropolsky, A.M. Samoilenko and 0.1. Martinyuk: Systems of Evolution Equations with Periodic and Quasiperiodic Coefficients. 1992,280 pp.
ISBN 0-7923-2054-9
I.T. Kiguradze and T.A. Chanturia: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. 1992, 332 pp. ISBN 0-7923-2059-X
V.L. Kocic and G. Ladas: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. 1993, 228 pp. ISBN 0-7923-2286-X
S. Levendorskii: Degenerate Elliptic Equations. 1993,445 pp. ISBN 0-7923-2305-X
D. Mitrinovic and J.D. Keckic: The Cauchy Method of Residues, Volume 2. Theory and Applications. 1993, 202 pp. ISBN 0-7923-2311-8
R.P. Agarwal and PJ.Y Wong: Error Inequalities in Polynomial Interpolation and Their Applications. 1993,376 pp. ISBN 0-7923-2337-8
A.G. Butkovskiy and L.M. Pustyl'nikov (eds.): Characteristics of DistributedParameter Systems. 1993,386 pp. ISBN 0-7923-2499-4
B. Steroin and V. Shatalov: Differential Equations on Complex Manifolds. 1994, 504 pp. ISBN 0-7923-2710-1
S.B. Yakubovich and Y.F. Luchko: The Hypergeometric Approach to Integral Transforms and Convolutions. 1994,324 pp. ISBN 0-7923-2856-6
C. Gu, X. Ding and C.-C. Yang: Partial Differential Equations in China. 1994, 181 pp. ISBN 0-7923-2857-4
V.G. Kravchenko and G.S. Litvinchuk: Introduction to the Theory of Singular Integral Operators with Shift. 1994,288 pp. ISBN 0-7923-2864-7
A. Cuyt (ed.): Nonlinear Numerical Methods and Rational Approximation 11. 1994, 446 pp. ISBN 0-7923-2967-8
Other Mathematics and Its Applications titles of interest:
G. Gaeta: Nonlinear Symmetries and Nonlinear Equations. 1994, 258 pp. ISBN 0-7923-3048-X
V.A. Vassiliev: Ramified Integrals, Singularities and Lacunas. 1995,289 pp. ISBN 0-7923-3193-1
NJa. Vilenkin and A.U. Klimyk: Representation of Lie Groups and Special Functions. Recent Advances. 1995,497 pp. ISBN 0-7923-3210-5
Yu. A. Mitropolsky and A.K. Lopatin: Nonlinear Mechanics, Groups and Symmetry. 1995,388 pp. ISBN 0-7923-3339-X
R.P. Agarwal and P.Y.H. Pang: Opial Inequalities with Applications in Differential and Difference Equations. 1995,393 pp. ISBN 0-7923-3365-9
A.G. Kusraev and S.S. Kutateladze: Subdifferentials: Theory and Applications. 1995,408 pp. ISBN 0-7923-3389-6
M. Cheng, D.-G. Deng, S. Gong and c.-c. Yang (eds.): Harmonic Analysis in China. 1995,318 pp. ISBN 0-7923-3566-X
M.S. Livsic, N. Kravitsky, A.S. Markus and V. Vinnikov: Theory of Commuting Nonselfadjoint Operators. 1995, 314 pp. ISBN 0-7923-3588-0
A.I. Stepanets: Classification and Approximation of Periodic Functions. 1995,360 pp. ISBN 0-7923-3603-8
c.-G. Ambrozie and F.-H. Vasilescu: Banach Space Complexes. 1995,205 pp. ISBN 0-7923-3630-5
E. Pap: Null-Additive Set Functions. 1995,312 pp. ISBN 0-7923-3658-5
C.J. Colboum and E.S. Mahmoodian (eds.): Combinatorics Advances. 1995, 338 pp. ISBN 0-7923-3574-0
V.G. Danilov, V.P. Maslov and K.A. Volosov: Mathematical Modelling of Heat and Mass Transfer Processes. 1995,330 pp. ISBN 0-7923-3789-1
A. LaurinCikas: Limit Theorems for the Riemann Zeta-Function. 1996, 312 pp. ISBN 0-7923-3824-3
A. Kuzhel: Characteristic Functions and Models of Nonself-Adjoint Operators. 1996,283 pp. ISBN 0-7923-3879-0
G.A. Leonov, I.M. Burkin and A.I. Shepeljavyi: Frequency Methods in Oscillation Theory. 1996,415 pp. ISBN 0-7923-3896-0
B. Li, S. Wang, S. Yan and c.-c. Yang (eds.): Functional Analysis in China. 1996, 390 pp. ISBN 0-7923-3880-4
P.S. Landa: Nonlinear Oscillations and Waves in Dynamical Systems. 1996, 554 pp. ISBN 0-7923-3931-2
Other Mathematics and Its Applications titles of interest:
A.J. Jerri: Linear Difference Equations with Discrete Transform Methods. 1996, 462 pp. ISBN 0-7923-3940-1
I. Novikov and E. Semenov: Haar Series and Linear Operators. 1997,234 pp. ISBN 0-7923-4006-X
L. Zhizhiashvili: Trigonometric F ourier Series and Their Conjugates. 1996, 312 pp. ISBN 0-7923-4088-4
R.G. Buschman: Integral Transformation, Operational Calculus, and Generalized Functions. 1996,246 pp. ISBN 0-7923-4183-X
V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan: Dynamic Systems on Measure Chains. 1996,296 pp. ISBN 0-7923-4116-3
D. Guo, V. Lakshmikantham and X. Liu: Nonlinear Integral Equations in Abstract Spaces. 1996,350 pp. ISBN 0-7923-4144-9
Y. Roitberg: Elliptic Boundary Value Problems in the Spaces of Distributions. 1996, 427 pp. ISBN 0-7923-4303-4
Y. Komatu: Distortion Theorems in Relation to Linear Integral Operators. 1996, 313 pp. ISBN 0-7923-4304-2
A.G. Chentsov: Asymptotic Attainability. 1997,336 pp. ISBN 0-7923-4302-6
S.T. Zavalishchin and A.N. Sesekin: Dynamic Impulse Systems. Theory and Applications. 1997, 268 pp. ISBN 0-7923-4394-8
U. Elias: Oscillation Theory of Two-Term Differential Equations. 1997,226 pp. ISBN 0-7923-4447-2
D. O'Regan: Existence Theory for Nonlinear Ordinary Differential Equations. 1997, 204 pp. ISBN 0-7923-4511-8
Yu. Mitropo1skii, G. Khoma and M. Gromyak: Asymptotic Methods for Investigating Quasiwave Equations of Hyperbolic Type. 1997,418 pp. ISBN 0-7923-4529-0
R.P. Agarwal and PJ.Y. Wong: Advanced Topics in Difference Equations. 1997, 518 pp. ISBN 0-7923-4521-5
N.N. Tarkhanov: The Analysis of Solutions of Elliptic Equations. 1997, 406 pp. ISBN 0-7923-4531-2
B. Rieean and T. Neubrunn: Integral, Measure, and Ordering. 1997,376 pp. ISBN 0-7923-4566-5
N.L. Gol' dman: Inverse Stefan Problems. 1997, 258 pp. ISBN 0-7923-4588-6
S. Singh, B. Watson and P. Srivastava: Fixed Point Theory and Best Approximation: The KKM-map Principle. 1997,230 pp. ISBN 0-7923-4758-7
A. Pankov: G-Convergence and Homogenization of Nonlinear Partial Differential Operators. 1997,263 pp. ISBN 0-7923-4720-X
Other Mathematics and Its Applications titles of interest:
S. Hu and N.S. Papageorgiou: Handbook of Multivalued Analysis. Volume I: Theory. 1997,980 pp. ISBN 0-7923-4682-3 (Set of2 volumes: 0-7923-4683-1)
L.A. Sakhnovich: Interpolation Theory and Its Applications. 1997,216 pp. ISBN 0-7923-4830-0
G.V. Milovanovic: Recent Progress in Inequalities. 1998,531 pp. ISBN 0-7923-4845-1
V.V. Filippov: Basic Topological Structures of Ordinary Differential Equations. 1998,530 pp. ISBN 0-7293-4951-2
S. Gong: Convex and Starlike Mappings in Several Complex Variables. 1998, 208 pp. ISBN 0-7923-4964-4
A.B. Kharazishvili: Applications of Point Set Theory in Real Analysis. 1998, 244 pp. ISBN 0-7923-4979-2
R.P. Agarwal: Focal Boundary Value Problems for Differential and Difference Equations. 1998,300 pp. ISBN 0-7923-4978-4
D. Przeworska-Rolewicz: Logarithms and Antilogarithms. An Algebraic Analysis Approach. 1998,358 pp. ISBN 0-7923-4974-1
Yu. M. Berezansky and A.A. Kalyuzbnyi: Harmonic Analysis in Hypercomplex Systems. 1998,493 pp. ISBN 0-7923-5029-4
V. Lakshmikantham and A.S. Vatsala: Generalized Quasilinearization for Nonlinear Problems. 1998,286 pp. ISBN 0-7923-5038-3
V. Barbu: Partial Differential Equations and Boundary Value Problems. 1998, 292 pp. ISBN 0-7923-5056-1
J. P. Boyd: Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics. Generalized Solitons and Hyperasymptotic Perturbation Theory. 1998, 61O-pp.
ISBN 0-7923-5072-3