A class of doubly periodic waves for nonlinear evolution equations

Wave Motion 35 (2002) 71–90

A class of doubly periodic waves for nonlinear evolution equations

K.W. Chow∗Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong

Received 15 November 1999; received in revised form 23 August 2000; accepted 9 January 2001

Abstract

A class of doubly periodic waves for several nonlinear evolution equations is studied by the Hirota bilinear method.Analytically these waves can be expressed as rational functions of elliptic functions with different moduli, and may correspondto standing as well as propagating waves. The two moduli are related by a condition determined as part of the solution process,and the condition translates into constraints on the wavenumbers allowed. Such solutions for the nonlinear Schrödingerequation agree with results derived earlier in the literature by a different method. The present method of combining the Hirotamethod, elliptic and theta functions is applicable to a wider class of equations, e.g., the Davey–Stewartson, the sinh-Poissonand the higher dimensional sine-Gordon equations. A long wave limit is studied for these special doubly periodic solutionsof the Davey–Stewartson and Kadomtsev–Petviashvili equations, and the results are the generation of new solutions and theemergence of the component solitons as the fundamental building blocks, respectively. The validity of these doubly periodicsolutions is verified by MATHEMATICA. © 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

Solitary and periodic waves of nonlinear evolution equations (NEEs) have been studied intensively. Relevantfields include biology, hydrodynamics, optics, plasma physics and solid state physics. Examples of such integrableequations are the nonlinear Schrödinger, Davey–Stewartson and sine-Gordon equations [1]. The goal of the presentwork is to document a class of doubly periodic solutions for the several typical and important members in this vastfamily of special evolution equations. These new entities may correspond to standing as well as propagating waves.

Various forms of periodic waves for these NEEs are known in the literature. The most obvious and elementaryclass comprises of a single Jacobi elliptic or genus one theta function [2]. Another type consists of rational functionsof trigonometric and hyperbolic functions [3,4]. Still another class is constructed in terms of rational functionsof elliptic/theta functions of different moduli [5,6]. Many ingenious and elegant techniques have been invented orgeneralized in treating nonlinear waves, e.g., the inverse scattering transform, Bäcklund transformation and Hirotabilinear method. The Hirota method [7] and theta functions will be the primary tools employed in the present work.

The nonlinear Schrödinger equation (NLS±) is an important representative of envelope equations [1]:

i∂A

∂t+ ∂

2A

∂x2+ 2A2A∗ = 0, (1.1)

∗ Tel.: +852-2859-2641; fax: +852-2858-5415.E-mail address: [email protected] (K.W. Chow).

0165-2125/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0165 -2125 (01 )00078 -6

72 K.W. Chow / Wave Motion 35 (2002) 71–90

i∂A

∂t+ ∂

2A

∂x2− 2A2A∗ = 0, (1.2)

where A∗ denotes the complex conjugate of A.It governs the evolution of a weakly nonlinear wave packet in fluids, optics, and plasma physics. In the context

of nonlinear fiber optics, (1.1) and (1.2) describe the anomalous (normal) dispersion regime; bright (dark) solitonsoccur and the plane wave shows modulational instability (stability).

The first task here is to examine a class of solutions for NLS+ (1.1) and NLS− (1.2), and the result will providemotivation for a similar search for other NEEs. To be precise, NLS− possesses a doubly periodic solution in termsof the classical Jacobi elliptic functions:

A = rk1√1 + k1

[cn(st, k1)+ i

√1 + k1sn(st, k1)dn(rx, k)√

1 + k1dn(rx, k)+ dn(st, k1)

]exp

(− 2ir2

1 + k1t

), (1.3)

s = 2r2

1 + k1, k2 = 2k1

1 + k1. (1.4)

This solution was derived by a special ansatz relating to the order of solution applied to NLS− [5]. This methodinvolves the reduction of the original partial differential equation, which has an infinite number of degrees offreedom, to a dynamical system with a finite number of degrees of freedom. Both solitary waves and periodicsolutions are obtained by this special ansatz, and hence this procedure provides a deeper understanding on therelationship between localized and periodic wave solutions.

In this paper we first rederive the same doubly periodic solution for NLS+ and NLS− using the Hirota operatorand theta functions. We then use the same reasoning and technique to derive similar doubly periodic solutions forother evolution equations, where the special ansatz or the procedure of reduction of degrees of freedom might beimpossible, not available or simply not yet investigated completely. The advantage of the Hirota approach is thatmany integrable evolution equations have Hirota forms.

The structure of this paper can now be elucidated. In this section the existing developments for NLS− arereviewed and the notations for theta functions are defined. In Section 2 results of an extended search for NLS+employing theta functions and the Hirota operator are reported. They are consistent with those discovered recentlywith the ‘order of solution’ method [8]. With this assurance the powerful conjunction of theta functions and bilinearmethod is applied to other significant NEEs where the special ansatz is unavailable in the literature. In Section3 a higher order NLS which admits N bright solitons is treated [9]. Doubly periodic solutions similar to those ofNLS+ are found. In Section 4 the Davey–Stewartson equation, a key system of (2 + 1)-dimensional (two spatialand one temporal) NEEs is investigated [1]. Different families of doubly periodic standing waves are obtained,depending on the relative magnitude of the wavenumbers in the x, y directions. In Section 5 the sinh-Poissonequation, which has relevance in inviscid vortex dynamics and Navier–Stokes turbulence in bounded domain, isstudied [10–12]. In Section 6 we focus on a (2 + 1)-dimensional generalization of the sine-Gordon model [13].The system possesses kinks, breathers and other special solutions. We shall show that doubly periodic solutionsalso exist. A long wave limit for some typical doubly periodic waves is taken in Section 7. This results in stillanother class of solutions not yet fully documented in the literature. Some solutions are periodic in one directiononly.

An alternative representation for the doubly periodic solution of NLS− (1.3) and (1.4) in terms of theta functions(Appendix A) is [6]

A= αθ3(0, τ )θ4(0, τ )θ2(0, τ1)θ4(0, τ1)

[θ2(ωt, τ1)θ4(αx, τ )+ iθ1(ωt, τ1)θ3(αx, τ )

θ3(ωt, τ1)θ4(αx, τ )+ θ4(ωt, τ1)θ3(αx, τ )]

exp(−iΩt),

ω= 2α2θ23 (0, τ )θ

24 (0, τ )

θ24 (0, τ1)

, Ω = α2(θ43 (0, τ )+ θ4

4 (0, τ )),θ2

3 (0, τ1)

θ24 (0, τ1)

= θ43 (0, τ )+ θ4

4 (0, τ )

2θ23 (0, τ )θ

24 (0, τ )

. (1.5)

K.W. Chow / Wave Motion 35 (2002) 71–90 73

The advantages of employing theta functions are

1. the representations of these new exact solutions are more symmetric,2. the Hirota bilinear approach can be utilized and a systematic search is feasible, and3. this method can be readily generalized to other NEEs which possess bilinear forms.

The periodic solutions of several classical integrable equations, e.g., the sine-Gordon and the Kadomtsev–Petviashvili equations have received elaborate and elegant treatments in terms of the geometry of Riemann surfaces[15]. We wish to assert that the Hirota approach provides a reasonable alternative, and can be readily extended toother NEEs. The sinh-Poisson equation will be selected as an illustrative example for the purpose of discussion.

Although a general formulation of doubly (in fact multiply) periodic solutions of the sine-Gordon and sinh-Poissontype equations has been given earlier in the literature [16], our results relating the two different moduli of the ellipticfunctions (Section 5) are purely analytical and do not require extensive knowledge of geometry and scatteringtheory. The wave numbers are related by relatively simple expressions and no numerical inversion is necessary.Furthermore, only genus one theta functions or ordinary Jacobi elliptic functions are involved, and their computationsand graphic display can be handled by most available computer algebra packages. The search process can thus bereadily generalized for other evolution equations possessing bilinear forms.

2. The NLS+

The motivation of the present study was to derive solutions similar to (1.3)–(1.5) for NLS+. In earlier works thechoice of θ3, θ4 for the denominator of the NLS− solutions (1.3)–(1.5) is natural since they do not have real zeros. Inorder to solve NLS+ or to locate further families of rational expressions of elliptic functions, a more general and flexi-ble choice for the theta functions must be made. In particular individual theta functions with real zeros (θ1, θ2)must beincluded in the consideration, but the final result must be adjusted or checked to be free of singularities. The remainingcomponents must then be computed so that the relevant Hirota bilinear equations are satisfied. Additional constraintson the moduli (nomes) of the elliptic (theta) functions usually arise and their physical significance will be examined.

To be precise one starts with the Hirota bilinear equation of NLS+ (1.1)

A = g

f, (iDt + D2

x − C)g·f = 0, (D2x − C)ff = 2gg∗, (2.1)

where f is real, C is a constant and D the Hirota operator. In subsequent usage C will denote a generic constant inthe bilinear form of NEEs, and the value will vary with individual cases.

If we choose

f = θ4(ωt, τ1)θ4(αx, τ )− θ2(ωt, τ1)θ2(αx, τ ), (2.2)

we obtain the exact solution

A = αθ2(0, τ )θ4(0, τ )θ3(0, τ1)

θ4(0, τ1)


θ4(ωt, τ1)θ4(αx, τ )− θ2(ωt, τ1)θ2(αx, τ )]

exp(−iΩt), (2.3)

θ44 (0, τ )− θ4

2 (0, τ )

2θ22 (0, τ )θ

24 (0, τ )

= θ22 (0, τ1)

θ24 (0, τ1)

, ω = 2α2θ22 (0, τ )θ

24 (0, τ )

θ24 (0, τ1)

, Ω = α2(θ42 (0, τ )− θ4

4 (0, τ )). (2.4)

The Hirota derivatives of theta functions are handled by a collection of identities. The technical details are describedin Appendix A.

The elliptic function counterpart for (2.3) and (2.4) is

A = r√2

[dn(st, k1)cn(rx, k)/(

√1 + k1)+ i

√k1 sn(st, k1)

1 − √k1 cn(st, k1)cn(rx, k)/

√1 + k1

]exp(−iΩt) (2.5)


with

k1 = 1 − 2k2 > 0, s = r2, Ω = r2(2k2 − 1) = −r2k1. (2.6)

The sign in (2.2) is immaterial due to Galilean invariance, θ2(x + π) = −θ2(x). We start with the minus sign toreproduce the known solutions (2.5) and (2.6) in a more convenient manner.

Inconsistency will occur if one uses θ1 instead of θ2 in (2.2). If one has insisted on using θ3 instead of θ2 in(2.2) one would obtain a purely imaginary wavenumber. The Jacobi imaginary transformation in the theory of thetafunctions is now invoked, and another exact solution for NLS+ can then be deduced:

A = αθ2(0, τ )θ3(0, τ )θ2(0, τ1)

θ4(0, τ1)



exp(−iΩt),

θ42 (0, τ )+ θ4

3 (0, τ )

2θ22 (0, τ )θ

23 (0, τ )

= θ23 (0, τ1)

θ24 (0, τ1)

, ω = 2α2θ22 (0, τ )θ

23 (0, τ )

θ24 (0, τ1)

, Ω = −α2(θ42 (0, τ )+ θ4

3 (0, τ )).

The elliptic function version is

A = rk1√1 + k1

[cn(st, k1) cn(rx, k)+ i

√1 + k1 sn(st, k1) dn(rx, k)√

1 + k1 dn(rx, k)− dn(st, k1) cn(rx, k)

]exp

(2ir2t

1 + k1

),

k =√

1 − k1

1 + k1, s = 2r2

1 + k1.

This again agrees with results obtained by the ansatz method [8]. However, in this paper we shall focus on thesearch that does not require the Jacobi imaginary transformation. Results on exact solutions that do require suchtransformation will be reported in a separate work.

3. The Hirota equation — a special higher order NLS

NLS is typically only the leading order approximation in the evolution of the wave packet. Higher order terms arenecessary when additional physics is needed. In hydrodynamics, surface gravity waves of moderate steepness callfor these fourth-order NEEs. In nonlinear fiber optics, studies of pulses of short duration (the femtosecond regime)also demand the introduction of these higher order terms. Although such higher order NLS equations are generallynot integrable, a special version, the Hirota equation [9],

iAt + Axx + 2A2A∗ + ivAxxx + 6ivAA∗Ax = 0, (3.1)

admits N (bright) soliton solutions. We now attempt to search for doubly periodic solutions of (3.1) to test therobust nature of this technique. The dark soliton regime of the Hirota equation is considered earlier in the literaturein conjunction with NLS−.

Expressions for doubly periodic standing waves will now be derived from the bilinear form of (3.1)

A = G

f, (D2

x − C)ff = 2GG∗, (iDt − 3ivCDx + D2x + ivD3

x − C)G·f = 0. (3.2)

We now further introduce

G = g exp[i(px −Ωt)], (D2x − C)ff = 2gg∗, (3.3)

[iDt+i(−3vC + 2p − 3vp2)Dx + (1 − 3vp)D2x + ivD3

x + (Ω + 3vCp − p2 + vp3 − C)]g·f = 0. (3.4)


For brevity only one family of solutions mentioned in Section 2 will be shown, and the other can be obtained in asimilar fashion. We again start with the choice

f = θ4(ωt, τ1)θ4(αx, τ )− θ2(ωt, τ1)θ2(αx, τ ). (3.5)

A remark on the procedure applicable to all the new solutions in this paper is in order. Basically we assume theform of the denominator in the Hirota transformation to be

f = θ4(αx, τ )θ4(ωt, τ1)+ θm(αx, τ )θm(ωt, τ1), (3.6)

and ωt might be replaced by βy in problems involving two spatial dimensions. The case m = 3 has been treatedearlier in the literature [6]. The present work addresses the situation for m = 1 or 2. The precise form for thenumerator must be determined from the governing bilinear equations under consideration, using the formulas forthe Hirota derivatives of theta functions (Appendix A). The constraint on the two parameters τ, τ1 in the finalsolution must be used to check that (3.5) does not vanish.

For the present example the structure of the bilinear equations (3.3) and (3.4) shows that consistency is achievedonly if the contributions from the Dx and D3

x terms sum up to zero. One restriction on p must be imposed:

p2 − 2p

3v+ 2α2

3(θ4

2 (0, τ )− θ44 (0, τ )) = 0, (3.7)

but this equation always has real roots as the constant term in (3.7) is always negative (3.8). This scenario is quiteunlike that of the dark soliton regime.

An exact solution for (3.1) is thus

A= αθ2(0, τ )θ4(0, τ )θ3(0, τ1)θ4(0, τ1)



exp(i(px −Ωt)),

θ44 (0, τ )− θ4

2 (0, τ )

2θ22 (0, τ )θ

24 (0, τ )

= θ22 (0, τ1)

θ24 (0, τ1)

,

ω= 2α2(1 − 3vp)θ22 (0, τ )θ

24 (0, τ )

θ24 (0, τ1)

, Ω = (1 − 3vp)α2(θ42 (0, τ )− θ4

4 (0, τ ))+ p2 − vp3. (3.8)

Doubly periodic solutions are thus derived for this higher order NLS equation too, provided that the constraintk < 1/

√2 holds (2.6).

4. The Davey–Stewartson equations

The Benney–Roskes, Davey–Stewartson (DS) equations constitute the governing equations for the evolution of(2 + 1)-dimensional (two spatial and one temporal) weakly nonlinear wave packets in hydrodynamics [1]:

iAt − Axx − σ 2Ayy + vA2A∗ = 2QA, Qxx − σ 2Qyy = v(AA∗)xx. (4.1)

Theoretically DS play a fundamental role in the theory of solvable (2 + 1)-dimensional NEEs. σ = 1(σ = i)corresponds to the DSI (DSII) case. The bilinear form is (A = G/f,Q = 2(log f )xx, f = real, C = constant)

(iDt − D2x − σ 2D2

y − C)G·f = 0, (D2x − σ 2D2

y − C)ff = vGG∗. (4.2)

“Doubly periodic” here implies that the solution is periodic in both the x, y directions. Intuitively a theta function ofgenus two is required to describe these doubly periodic waves. The surprising point is that, at least in the particularcase considered here, a product of two genus one functions is sufficient. This is of great practical as well as academic


interest, since genus one functions greatly reduce the ensuing analytical and computational complexity. Travellingwaves have been studied extensively in the literature and the focus of the present section is on standing waves.Indeed we shall derive below several different families of doubly periodic standing waves, for DSI as well as forDSII, using rational expressions of genus one theta functions.

The central themes of the present work, namely, theta functions and the bilinear transformation, will again beutilized. A search based on the symmetry of theta function identities enables us to deduce the following exactsolutions.

(A) DSI(σ = 1, v = 2):The dispersion relation and the parameters of one new exact solution are

A= λ1

[θ4(αx, τ )θ1(βy, τ1)+ θ1(αx, τ )θ4(βy, τ1)θ4(αx, τ )θ4(βy, τ1)+ θ1(αx, τ )θ1(βy, τ1)

]exp(−iΩt),

α[θ23 (0, τ )− θ2

2 (0, τ )] = β[θ23 (0, τ1)− θ2

2 (0, τ1)], λ21 = β2θ2

2 (0, τ1)θ23 (0, τ1)− α2θ2

2 (0, τ )θ23 (0, τ ),

Ω = α2[θ ′′

2 (0, τ )

θ2(0, τ )+ θ

′′3 (0, τ )

θ3(0, τ )+ 2θ ′′

4 (0, τ )

θ4(0, τ )

]− 2β2θ2

2 (0, τ1)θ23 (0, τ1), (4.3)

Q= R1

[θ4(αx, τ )θ4(βy, τ1)+ θ1(αx, τ )θ1(βy, τ1)]2,

R1 = α2θ24 (βy, τ1)

[2θ ′′

4 (0, τ )θ24 (αx, τ )

θ4(0, τ )− 2θ2

2 (0, τ )θ23 (0, τ )θ

21 (αx, τ )

]

+α2θ21 (βy, τ1)

[2θ ′′

4 (0, τ )θ21 (αx, τ )

θ4(0, τ )− 2θ2

2 (0, τ )θ23 (0, τ )θ

24 (αx, τ )

]

+2α2(θ ′′

2 (0, τ )

θ2(0, τ )+ θ

′′3 (0, τ )

θ3(0, τ )

)θ1(αx, τ )θ1(βy, τ1)θ4(αx, τ )θ4(βy, τ1). (4.4)

The validity of the new solutions is now verified independently by direct differentiation with a computer algebrasoftware. It will be necessary to first transform the theta formulation into the Jacobi elliptic functions. The argumentsof the theta and elliptic functions differ by the scalar θ2

3 (0), and the new wave numbers r, s are used.

r = αθ23 (0, τ ), s = βθ2

3 (0, τ1), k = θ22 (0, τ )

θ23 (0, τ )

, k1 = θ22 (0, τ1)

θ23 (0, τ1)

. (4.5)

A = λ1

(S1 + S1 + S1S

)exp(−iΩt), S =

√k sn(rx, k), S1 =

√k1 sn(sy, k1), (4.6)

Q = 2r2(

1 − E

K

)− 2r2

(k(S2

1 + S2)+ (1 + k2)S1S

(1 + S1S)2

), (4.7)

where K and E are complete elliptic integrals of the first and second kind, respectively,

K =∫ π/2

0

dξ√1 − k2 sin2ξ

, E =∫ π/2

0

√1 − k2 sin2ξ dξ

with

r(1 − k) = s(1 − k1), λ21 = −r2k + s2k1 = r2(1 − k)2

(k1

(1 − k1)2− k

(1 − k)2), (4.8)


Fig. 1. Intensity |A|2 of the complex envelope of DSI, r < s (4.6), k1 = 0.5, k = 0.4, r = 1.

Ω = 4r2(

1 − E

K

)− r2(1 + k2)− 2s2k1. (4.9)

The symbolic manipulation software MATHEMATICA is now used to verify that (4.6)–(4.9) indeed solve (4.1).A precaution in the actual implementation must be noted. Although k is used to denote the modulus here, Jacobielliptic functions in MATHEMATICA require an input ofm = k2. |A|2 is illustrated in Fig. 1. The restriction λ2

1 > 0implies k1 > k. Eq. (4.8) implies that the wavenumber in the x direction is smaller than that in the y direction. Theamplitude reduces to zero in the degenerate case of r = s or k = k1.

(B) DSI(σ = 1, v = 2): A related but not identical solution is

A = λ2

[θ4(αx, τ )θ1(βy, τ1)− θ1(αx, τ )θ4(βy, τ1)θ4(αx, τ )θ4(βy, τ1)+ θ1(αx, τ )θ1(βy, τ1)

]exp(−iΩt), (4.10)

whereQ is still given by (4.4) and (4.7), but the parameters are now related by

α[θ23 (0, τ )+ θ2

2 (0, τ )] = β[θ23 (0, τ1)+ θ2

2 (0, τ1)], λ22 = β2θ2

2 (0, τ1)θ23 (0, τ1)− α2θ2

2 (0, τ )θ23 (0, τ ),

Ω = α2[θ ′′

2 (0, τ )

θ2(0, τ )+ θ

′′3 (0, τ )

θ3(0, τ )+ 2θ ′′

4 (0, τ )

θ4(0, τ )

]+ 2β2θ2

2 (0, τ1)θ23 (0, τ1).


A = λ2

(S1 − S1 + S1S

)exp(−iΩt), S =

√k sn(rx, k), S1 =

√k1 sn(sy, k1), (4.11)


Fig. 2. Intensity |A|2 of the complex envelope of DSI, r > s (4.11), k1 = 0.7, k = 0.4, r = 1.

r(1 + k) = s(1 + k1), λ22 = −r2k + s2k1 = r2(1 + k)2

(k1

(1 + k1)2− k

(1 + k)2), (4.12)

Ω = 4r2(

1 − E

K

)− r2(1 + k2)+ 2s2k1. (4.13)

Again the restriction k1 > k is needed. Expression (4.12) in turn implies that the wavenumber in the x direction islarger than that in the y direction. Hence a long wave limit (k1 → 1) can be taken. Details are pursued in Section7. The validity of (4.10)–(4.13) is verified by MATHEMATICA. Fig. 2 illustrates |A|2.

(C) DSII(σ = i, v = 2): For DSII we start the search with

f = θ4(αx, τ )θ4(βy, τ1)+ θ2(αx, τ )θ2(βy, τ1). (4.14)

We now compute the form of g necessary to satisfy bilinear equations (4.2). The dispersion relation and the relevantparameters of one new exact solution for (4.1) are

A= λ3

[θ1(αx, τ )θ1(βy, τ1)+ iθ3(αx, τ )θ3(βy, τ1)

θ4(αx, τ )θ4(βy, τ1)+ θ2(αx, τ )θ2(βy, τ1)]

exp(−iΩt), αθ23 (0, τ ) = βθ2

3 (0, τ1),

λ23 = α2(θ2

4 (0, τ )θ24 (0, τ1)− θ2

2 (0, τ )θ22 (0, τ1))

θ23 (0, τ )

θ23 (0, τ1)

,

Ω = α2[θ ′′

3 (0, τ )

θ3(0, τ )+ θ

′′4 (0, τ )

θ4(0, τ )+ 2θ ′′

2 (0, τ )

θ2(0, τ )

]+ β2θ4

2 (0, τ1), (4.15)


Q= R2

[θ4(αx, τ )θ4(βy, τ1)+ θ2(αx, τ )θ2(βy, τ1)]2,

R2 = α2θ24 (βy, τ1)

[2θ2

2 (0, τ )θ24 (0, τ )θ

22 (αx, τ )+

2θ ′′3 (0, τ )θ

24 (αx, τ )

θ3(0, τ )

]

+α2θ22 (βy, τ1)

[2θ ′′

3 (0, τ )θ22 (αx, τ )

θ3(0, τ )− 2θ2

2 (0, τ )θ24 (0, τ )θ

24 (αx, τ )

]

+2α2(θ ′′

2 (0, τ )

θ2(0, τ )+ θ

′′4 (0, τ )

θ4(0, τ )

)θ2(αx, τ )θ4(αx, τ )θ2(βy, τ1)θ4(βy, τ1). (4.16)

The elliptic function version of (4.15) and (4.16) is (r = αθ23 (0, τ ), s = βθ2

3 (0, τ1))

A = λ3

(S1S + iD1D

1 + C1C

)exp(−iΩt), S =

√k sn(rx, k), S1 =

√k1 sn(ry, k1), (4.17)

C =√k cn(rx, k)

(1 − k2)1/4, C1 =

√k1 cn(ry, k1)

(1 − k21)

1/4, D = dn(rx, k)

(1 − k2)1/4, D1 = dn(ry, k1)

(1 − k21)

1/4, (4.18)

λ23 = r2(

√1 − k2

√1 − k2

1 − kk1), (4.19)

Ω = −4r2E

K+ r2(2 − k2)+ r2k2

1, (4.20)

Q = 2r2(

1 − E

K

)− 2r2k2 + 2r2

[k√

1 − k2(C2 − C21 )+ (2k2 − 1)C1C

(1 + C1C)2

], (4.21)

where C1 and C are given by (4.18). Eq. (4.19) implies the restriction k2 + k21 < 1.

The symbolic manipulation software MATHEMATICA is now used to verify that (4.17)–(4.21) indeed solve(4.1). Fig. 3 shows |A|2 and is generated by MATHEMATICA.

5. The sinh-Poisson equation

The sinh-Poisson equation,

ψxx + ψyy = − sinhψ, (5.1)

is relevant in inviscid, incompressible, two-dimensional steady fluid flows without body force, since the vorticity(ω) and the stream function (ψ) under those conditions are related by (f differentiable)

ψxx + ψyy = −ω = f (ψ). (5.2)

Novel flow patterns, e.g., a row of counter rotating vortices, constitute special exact solutions of (5.1) [11,12]. Theoccurrence of the sinh-Poisson equation in this area of fluid dynamics provides the motivation for studying theequation. We now focus on whether doubly periodic solutions exist for the sinh-Poisson equation [10].

The bilinear equations for (5.1) are

ψ = 4 tanh−1(g

f

)= 2 log

(f + gf − g

), (5.3)

(D2x + D2

y + C)(gg + ff) = 0, (D2x + D2

y + C + 1)g·f = 0. (5.4)


Fig. 3. Intensity |A|2 of the complex envelope of DSII (4.17), k1 = 0.5, k = 0.4, r = 1.

Expression (5.4) is different from the Hirota bilinear form of DS (4.2) in that the signs of D2x and D2

y are always thesame.

Following the spirit of the present paper, one now searches for exact solutions starting with

f = θ4(αx, τ )θ4(βy, τ1)+ θm(αx, τ )θm(βy, τ1).The choice m = 1 gives a new solution

ψ = 4 tanh−1[θ4(αx, τ )θ1(βy, τ1)− θ1(αx, τ )θ4(βy, τ1)θ4(αx, τ )θ4(βy, τ1)+ θ1(αx, τ )θ1(βy, τ1)

], (5.5)

β2[θ23 (0, τ1)− θ2

2 (0, τ1)]2 = 1 + 4α2θ2

2 (0, τ )θ23 (0, τ ),

α[θ22 (0, τ )+ θ2

3 (0, τ )] = β[θ22 (0, τ1)+ θ2

3 (0, τ1)].

The elliptic function counterpart is

ψ = 4 tanh−1

[ √k sn(rx, k)− √

k1 sn(sy, k1)

1 + √k√k1 sn(rx, k) sn(sy, k1)

], (5.6)

s2(1 − k1)2 = 1 + 4r2k, s(1 + k1) = r(1 + k). (5.7)


Fig. 4. Contour plot of the streamlines of the sinh-Poisson equation (5.6), k1 = 0.15, k = 0.15, r = 2.857, s = 2.857, light-positive,dark-negative.

Eq. (5.7) imposes the restriction

k1 <

(1 − √

k

1 + √k

)2

. (5.8)

Eq. (5.6) is always well defined as the absolute value of the expression is always less than 1 if (5.8) holds. Computeralgebra verification of (5.1)–(5.6) is performed. A contour plot generated by MATHEMATICA shows a rectangulararray of vortices of alternating signs (Fig. 4).

6. Two further examples

To illustrate the technique further we consider two more examples of purely real NEEs in two spatial dimensions.(A) The (2 + 1)-dimensional (two spatial and one temporal) extension of the sine-Gordon equation is(

θxt

sin θ

)x

−(θyt

sin θ

)y

−(θyθxt − θxθyt

sin2θ

)= 0. (6.1)


When the y dependence is absent (6.1) reduces to the (1 + 1)-dimensional sine-Gordon equation in characteristicscoordinates:

θxt = sin θ.

There is considerable evidence that a simple (2 + 1) extension of sine-Gordon equation, namely,

θxx + θyy − θtt = sin θ.

is not integrable. Hence (6.1) is receiving attention given the current interest in (2 + 1)-dimensional NEEs solvableby the inverse scattering transform [13]. Another remarkable motivation is that (6.1) arises in the theory of classicaldifferential geometry. There exists a gauge transform which maps (6.1) to the system:

uxyt + uxvyt + uyvxt = 0, vxy = uxuy. (6.2)

From an engineering viewpoint variations of (6.2) form a generalization of the real, pumped, Maxwell–Blochsystem, which has a physical interpretation in nonlinear optics. A vast family of solutions can be expressed in termsof the Painleve functions.

The goal of this section is to discover another class of solutions for (6.2) in terms of elliptic and theta functions.One starts with the bilinear form of (6.2):

u = i log

(G

F

)= −2 tan−1

(g

f

), G = F ∗ = f + ig, (6.3)

v = λxt + µyt+log(G·F) = λxt+µyt+log(g2 + f 2), (DxDyDt + λDy + µDx)G·F, DxDyG·F = 0,

(6.4)

(DxDyDt + λDy + µDx)g·f = 0, DxDy(ff + gg) = 0. (6.5)

We shall look for solutions of the form

f = θ4(αx − ωt, τ )θ4(βy −Ωt, τ1)+ θ1(αx − ωt, τ )θ1(βy −Ωt, τ1). (6.6)

The identity

DxDyDt [a(x − c1t)b(y − c2t) · c(x − c1t)d(y − c2t)] = −c1(D2xa · c)(Dyb · d)− c2(Dxa · d)(D2

yb · d),c1 = ω

α, c2 = Ω

β(6.7)

will be helpful in the intermediate calculations.A computation similar to previous sections yields the exact solution:

u = −2 tan−1[θ4(αx − ωt, τ )θ1(βy −Ωt, τ1)+ θ1(αx − ωt, τ )θ4(βy −Ωt, τ1)θ4(αx − ωt, τ )θ4(βy −Ωt, τ1)+ θ1(αx − ωt, τ )θ1(βy −Ωt, τ1)

], (6.8)

ω = λ

α[2θ ′′4 (0, τ )/θ4(0, τ )+ 2θ2

2 (0, τ )θ23 (0, τ )]

, Ω = µ

β[2θ ′′4 (0, τ1)/θ4(0, τ1)+ 2θ2

2 (0, τ1)θ23 (0, τ1)]

.


u = −2 tan−1

[ √k sn(r(x − c1t), k)+

√k1 sn(s(y − c2t), k1)

1 + √k√k1 sn(r(x − c1t), k)sn(s(y − c2t), k1)

], (6.9)


Fig. 5. u of the two-dimensional sine-Gordon system (6.9) at t = 0, r = 1, s = 1, k1 = 0.5, k = 0.5.

c1 = λ

2r2(1 + k − E/K), c2 = µ

2s2(1 + k1 − E1/K1), (6.10)

where E1,K1 are the complete elliptic integrals in terms of the modulus k1. Eq. (6.9) is illustrated in Fig. 5. Thedoubly periodic pattern moves in the x, y directions with speeds c1, c2 (6.10), respectively.

(B) The system

ut + uxxx + uyyy + 3uxwxx + 3uywyy − u3x − w3

y = 0, (6.11)

wxy = uxuy, (6.12)

passes the 3-soliton test and the bilinear Painleve test [17]. The bilinear form is

u = −2 tan−1(g

f

), w = log(g2 + f 2), (6.13)

(Dt + D3x + D3

y)gf = 0, DxDy(gg + ff) = 0. (6.14)

A doubly periodic, propagating wave is given by

f = θ4(αx − ωt, τ )θ4(βy −Ωt, τ1)+ θ3(αx − ωt, τ )θ3(βy −Ωt, τ1),g = θ4(αx − ωt, τ )θ3(βy −Ωt, τ1)+ θ3(αx − ωt, τ )θ4(βy −Ωt, τ1),


and (6.13). The phase speeds are

c1 = ω

α= α2

(θ ′′′

1 (0, τ )

θ ′1(0, τ )

+ 3θ ′′2 (0, τ )

θ2(0, τ )

), c2 = Ω

β= β2

(θ ′′′

1 (0, τ1)

θ ′1(0, τ1)

+ 3θ ′′2 (0, τ1)

θ2(0, τ1)

).

7. Long wave limit

The long wave limit of periodic waves will typically regenerate solitary and kink type wave forms. Such limitfor the present family of doubly periodic waves is especially rich, since one can proceed with the long wave limitin one direction only. No comprehensive study will be attempted but two cases, namely the Davey–Stewartson andthe Kadomtsev–Petviashvili equations, will be presented as illustrative examples.

(A) For DSI a long wave limit yields a sequence of ‘solitoffs’ [18]. A solitoff is a wave decaying in all directionsexcept a preferred one, or a semi-infinite solitary wave. For the DSI solutions (4.11)–(4.13) the limit k1 → 1 willyield

A = r(1 − k)2

[tanh sy − √

k sn(rx, k)

1 + √k sn(rx, k) tanh sy

]exp(−iΩt), (7.1)

Q = 2r2(

1 − E

K

)− 2r2

[k(k sn2(rx, k)+ tanh2sy)+ √

k(1 + k2)sn(rx, k)(tanh sy)

(1 + √k sn(rx, k) tanh sy)2

], (7.2)

s = r(1 + k)2

, Ω = 4r2(

1 − E

K

)− r

2(1 − k)22

. (7.3)

Fig. 6a shows the array of solitoffs. Computer algebra is used to verify that (7.1)–(7.3) solve (4.1).For DSII the limit k1 → 1, k → 0,

√k/(1 − k2

1)1/4 = m, 0 < m < 1, is taken for the solutions (4.17)–(4.21). A

new solution of DSII is obtained:

A =(r√

1 −m2 sech ry

1 +m cos rx sech ry

)exp(ir2t), (7.4)

Q = −2mr2 sech ry(cos rx +m sech ry)

(1 +m cos rx sech ry)2. (7.5)

This solution for A is periodic in the x direction and exponentially decaying in the y direction. Fig. 6b illustratesthe situation.

(B) For the Kadomtsev–Petviashvili equation the doubly periodic waves degenerate to the component 2-soliton.The Kadomtsev–Petviashvili (KP) equation,

(ut + 6uux + uxxx)x + σuyy = 0, (7.6)

governs the evolution of shallow water waves in (2 + 1) dimensions. Positive (negative) values of σ correspond tothe negative (positive) dispersion regime and the bilinear form is

u = 2(log f )xx, (DxDt + D4x + σD2

y − C)ff = 0. (7.7)

One doubly periodic solution is

f = θ3(αx − ωt, τ )θ4(βy, τ1)+ θ4(αx − ωt, τ )θ3(βy, τ1), (7.8)

u = R0

[θ3(αx − ωt, τ )θ4(βy, τ1)+ θ4(αx − ωt, τ )θ3(βy, τ1)]2, (7.9)


Fig. 6. (a) Intensity |A|2 of the long wave limit of DSI (7.1) k = 0.7, r = 1. (b) Intensity |A|2 of the long wave limit of DSII (7.4) r = 1,m = 0.5.


R0 = α2θ24 (βy, τ1)

[2θ ′′

2 (0, τ )θ23 (αx − ωt, τ )θ2(0, τ )

+ 2θ23 (0, τ )θ

24 (0, τ )θ

24 (αx − ωt, τ )

]

+α2θ23 (βy, τ1)

[2θ2

3 (0, τ )θ24 (0, τ )θ

23 (αx − ωt, τ )+ 2θ ′′

2 (0, τ )θ24 (αx − ωt, τ )θ2(0, τ )

]

+2α2(θ ′′

3 (0, τ )

θ3(0, τ )+ θ

′′4 (0, τ )

θ4(0, τ )

)θ3(αx − ωt, τ )θ3(βy, τ1)θ4(αx − ωt, τ )θ4(βy, τ1).


u = 2[log(X + Y )]xx + 2r2√

1 − k2X2 − 2r2E

K, X = dn(rx −Ωt, k)

(1 − k2)1/4, Y = dn(sy, k1)

(1 − k21)

1/4,

u = 2r2[√

1 − k2(1 −X4)+ (2 − k2)XY − 2√

1 − k2X3Y ]

(X + Y )2 + 2r2√

1 − k2X2 − 2r2E

K. (7.10)

The pattern propagates steadily in the x direction. The angular frequency Ω and the spanwise wavenumber s aregiven by

rΩ(2 − k2)− σs2(2 − k21) = r4

[−3k4 + 4(2 − k2)2 − 12(2 − k2)E

K

], (7.11)

rΩ√

1 − k2 − σs2√

1 − k21 = 4r4

√1 − k2

(2 − k2 − 3E

K

). (7.12)

A long wave limit can be taken by assuming√1 − k2 = ε,

√1 − k2

1 = mε, ε 1, m = 1.

Since dn(x)→ sech(x) as k → 1 the long wave limit is the 2-soliton for KP

u = 2r2

√m

[cosh(rx −Ωt) cosh sy + 1/

√m

(cosh sy + cosh(rx −Ωt)/√m)2], (7.13)

where

s =√

3

σ(1 −m)r2, Ω =

(4 −m1 −m

)r3.

This is a special case of and is similar to conclusions drawn by works earlier in the literature using the more elaboratetheory of genus two theta functions [15], but the calculations here are performed entirely using genus one thetafunctions. Fig. 7 shows the doubly periodic solution (7.10) and Fig. 8 illustrates the long wave limit (7.13).

Such consideration can be readily extended to other Korteweg–de Vries type bilinear equations where the geo-metric aspect has not been worked out completely, e.g., the (2 + 1)-dimensional Sadawa–Kotera equation

(D6x + 5DyD3

x − 5D2y + DxDt )ff = 0. (7.14)

Doubly periodic solutions can be sought by assuming f of the form (7.8), but details will not be pursued here.

8. Discussions and conclusions

A class of doubly periodic solutions of NEEs has been studied by employing the Hirota bilinear method, el-liptic and theta functions. Analytical expressions are obtained for the amplitude and frequencies in terms of the


Fig. 7. A doubly periodic solution of the Kadomtsev–Petviashvili equation (7.10), σ = 3, k = 0.25, r = 1.

wavenumbers and genus one theta constants of different moduli or nomes. For a few intensively studied NEEs,e.g., NLS, such approach and results are consistent with those derived from existing techniques, e.g., the geom-etry of Riemann surfaces or the ‘order of solution’ ansatz. The contribution here is to show that the Hirota andtheta combination can be applied to a wide variety of NEEs not treated before. In particular, if theta functionswith real zeros are included at the starting point of the search process, a much wider class of solutions can beobtained. The only necessary ingredient in the search is that the associated equations must possess bilinear forms.Examinations on the dispersion relations presented here show that it is often, but not always, necessary to imposeconstraints on the moduli of the elliptic functions involved. Those restrictions convert into extra conditions on thewavenumbers of these doubly periodic waves. Main results of the present work are summarized in the followingparagraphs.

1. Doubly periodic solutions for the NLS and a higher order NLS are obtained for the bright soliton regime, i.e.,the regime where the plane wave is unstable.

2. Classes of doubly periodic standing waves for the Davey–Stewartson I (DSI) equations with nodes (points of zerodisplacement) in the wave patterns are obtained. As an example of the moduli and wavenumbers restriction twodifferent families of solutions of DSI are derived, depending on the relative magnitude of the wavenumbers intwo mutually perpendicular directions. These families are examples of standing waves in two spatial dimensions.

3. A rectangular array of vortices of alternate signs in steady, inviscid, incompressible, two-dimensional flows isderived for the special case of sinh-Poisson vortex dynamics. Such flow patterns are indeed observed for fluidssubject to electromagnetic forces [19]. This case constitutes an example where the present technique can beapplied to a time independent problem.


Fig. 8. Long wave limit of the Kadomtsev–Petviashvili equation (7.13), σ = 3, r = 1.

4. Propagating waves in two spatial dimensions from the modified Korteweg–de Vries family of bilinear equations,e.g., the higher dimensional sine-Gordon equations, are derived to illustrate the applicability of the presentmethod.

5. A long wave limit for the doubly periodic solutions of this paper might yield yet another class of waves. For DSa sequence of periodic solitoffs is observed.

Currently intensive efforts are made to generalize this type of wave motion to coupled systems. In nonlinearoptics birefringence in fibers typically leads to coupled NLS. Travelling waves solutions are recently given for suchcoupled NLS in terms of the Weiestrass elliptic function [20]. It will be of great interest to examine if this currentclass of (standing waves like) solutions in terms of theta functions will be applicable to coupled NLS. The role ofx and t in (1.1) and (1.2) in optics might be different from hydrodynamics, and the physical interpretation must bemodified accordingly.

Imaginary wavenumbers might occasionally arise in the intermediate computations, and the classical Jacobiimaginary transformation of elliptic functions is then needed to deduce the correct wave form. Such situations haveonly been considered in this paper for NLS. Extension to other NEEs will be left for future studies.

Finally a remark regarding the theoretical aspect of NEEs is in order. A large number of bilinear equationsinvestigated to date falls into one of the four representative categories [21]. Classification of the equations in thiswork according to this scheme provides an instructive perspective:

1. NLS type: NLS and the Davey–Stewartson system ((2.1) and (4.2)),2. sine-Gordon type: the sinh-Poisson equation (5.4),


3. modified Korteweg–de Vries type: the (2 + 1)-dimensional sine-Gordon model (6.5),4. Korteweg–de Vries type: the Kadomtsev–Petviashvili and Sadawa–Kotera equations ((7.7) and (7.14)).

Hence the present theta formulation is quite universal in handling most bilinear equations commonly encountered.

Acknowledgements

The financial support of the Hong Kong Research Grants Council through contracts 7064/97E, 7067/98E,7066/00E and NSFC/HKU 8 is gratefully acknowledged. Mr. Derek W.C. Lai has kindly assisted in the productionof the figures.

Appendix A

The theta functions θn(x, τ ) [14], n = 1–4 and the parameters q (the nome), τ (pure imaginary) are defined by

θ1(x, τ ) = 2∞∑n=0

(−1)nq(n+1/2)2 sin(2n+ 1)x = −∞∑

m=−∞exp

(π iτ

(m+ 1

2

)2

+ 2i

(m+ 1

2

)(x + π

2

)),

θ2(x, τ ) = 2∞∑n=0

q(n+1/2)2 cos(2n+ 1)x =∞∑

m=−∞exp

(π iτ

(m+ 1

2

)2

+ 2i

(m+ 1

2

)x

),

θ3(x, τ ) = 1 + 2∞∑n=1

qn2

cos 2nx =∞∑

m=−∞exp(π iτm2 + 2imx),

θ4(x, τ ) = 1 + 2∞∑n=1

(−1)nqn2

cos 2nx =∞∑

m=−∞exp

(π iτm2 + 2im

(x + π

2

)),

0 < q < 1, q = exp(πiτ ), q = exp

(−πK

′

K

).

K , K ′ are the complete elliptic integrals of the first kind. Note the minus sign in front of θ1 ·θ1 is odd while theother three are even. The zeros of θ1, θ2, θ3, θ4 are atMπ + Nπτ , (M + 1

2 )π + Nπτ , (M + 12 )π + (N + 1

2 )πτ ,Mπ + (N + 1

2 )πτ , respectively, M and N are integers. Since θ1, θ2, (θ3, θ4) are related by a phase shift of π/2,there are roughly two groups of theta functions.

Some technical details of the intermediate calculations involved in this paper are provided. Relevant identities oftheta and elliptic functions are collected here.

sn(u) = θ3(0)θ1(z)

θ2(0)θ4(z), cn(u) = θ4(0)θ2(z)

θ2(0)θ4(z), dn(u) = θ4(0)θ3(z)

θ3(0)θ4(z),

z = u

θ23 (0)

, k = θ22 (0)

θ23 (0)

, k′ = θ24 (0)

θ23 (0)

, k2 + (k′)2 = 1.

Theta functions possess a huge number of identities, e.g.,

θ3(x + y)θ3(x − y)θ22 (0) = θ2

4 (x)θ21 (y)+ θ2

3 (x)θ22 (y), (A.1)


θ4(x + y)θ4(x − y)θ22 (0) = θ2

4 (x)θ22 (y)+ θ2

3 (x)θ21 (y). (A.2)

By considering the leading and quadratic terms in the Taylor series of y in identities of the form (A.1) and (A.2),one obtains

θ ′′4 (0)

θ4(0)− θ

′′3 (0)

θ3(0)= θ4

2 (0),θ ′′

4 (0)

θ4(0)− θ

′′2 (0)

θ2(0)= θ4

3 (0),θ ′′

3 (0)

θ3(0)− θ

′′2 (0)

θ2(0)= θ4

4 (0).

The Hirota operator and its properties are

Dmx Dnt g ·f =(∂

∂x− ∂

∂x′

)m (∂

∂t− ∂

∂t ′

)ng(x, t)f (x′, t ′)

∣∣∣∣x=x′,t=t ′

,

Dx(exp(imx)g ·exp(inx)f ) = [Dxgf + i(m− n)gf]exp(i(m+ n)x),D2x(exp(imx)g ·exp(inx)f ) = [D2

xgf + 2i(m− n)Dxgf − (m− n)2gf]exp(i(m+ n)x).Differentiating (A.1) and (A.2) with respect to y and setting y = 0 yield

D2xθ3(x)·θ3(x) = 2θ ′′

2 (0)θ23 (x)

θ2(0)+2θ2

3 (0)θ24 (0)θ

24 (x), D2

xθ4(x)·θ4(x) = 2θ23 (0)θ

24 (0)θ

23 (x)+

2θ ′′2 (0)θ

24 (x)

θ2(0),

Hence formulas for Dxθmθn, D2xθmθn can be developed for m, n integers using this line of reasoning.

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