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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 sech j (x) - 2j(j + 1)(j2 + 2j + 2) sechJ+2(x) +j(j + 1)(j + 2)(j + 3)sechj+4(x) (A.3) 482

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Page 1: IDENTITIES: TRIGONOMETRIC, SECH AND TANH - Springer978-1-4615-5825-5/1.pdf · IDENTITIES: TRIGONOMETRIC, SECH AND TANH "For me, the criterion of science is truth, but its motivation

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

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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

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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)

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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)

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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)

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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)

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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.

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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

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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)

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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)

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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)

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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)

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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

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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)

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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)

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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)

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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)

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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)

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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

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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).

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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

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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).

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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).

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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.)

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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)

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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.

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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

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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" .)

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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.

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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

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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

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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

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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

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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!.

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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

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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" .

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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-.

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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.

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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.

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Carrier, G. F. and Pearson, C. E.: 1968, Ordinary Differential Equations, Blais­dell, Waltham, MA. 229 pp.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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Finlayson, B. A.: 1973, The Method of Weighted Residuals and Variational Principles, Academic, New York. Pseudospectral method; 412 pp.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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572 WEAKLY NONLOCAL SOLITARY WAVES

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574 WEAKLY NONLOCAL SOLITARY WAVES

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578 WEAKLY NONLOCAL SOLITARY WAVES

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580 WEAKLY NONLOCAL SOLITARY WAVES

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solitons of the FKdV equation; demonstrates the coalescence of classical bions.

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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