Construction of Quasi-Periodic Wave Solutions for Differential- Difference Equation

Construction of Quasi-Periodic Wave Solutions for Differential- Difference Equation

Y. C. Hon^aand Qi Wang^b

aDepartment of Mathematics, Tat Chee Avenue 80, City University of Hong Kong, Hong Kong, PR China

bDepartment of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, PR China

Reprint requests to Q. W.; E-mail:wangqee@gmail.com

Z. Naturforsch.67a,21 – 28 (2012) / DOI: 10.5560/ZNA.2011-0063 Received August 25, 2011

Based on the use of the Hirota bilinear method and the Riemann theta function, we develop in this paper a constructive method for obtaining explicit quasi-periodic wave solutions of a new inte- grable generalized differential-difference equation. Analysis on the asymptotic property of the quasi- periodic wave solutions is given, and it is shown that the quasi-periodic wave solutions converge to the soliton solutions under certain conditions.

Key words:Hirota Bilinear Method; Riemann Theta Function; Quasi-Periodic Wave Solutions.

PACS numbers:03.65.Ge; 02.30.Ik

1. Introduction

Exact solutions of nonlinear equations have proven to be useful in simulating many real physical phenom- ena. The bilinear method developed by Hirota pro- vides a direct approach to construct exact solutions of nonlinear equations. In other words, if a nonlinear equation is written in bilinear forms by using an ap- propriate dependent variable transformation, its multi- soliton solutions can usually be obtained [1–9]. Re- cently, based on the Hirota bilinear method, Nakamura in his two serial papers [10,11] proposed a conve- nient way to construct a kind of quasi-periodic so- lutions of nonlinear equation from which the quasi- periodic wave solutions of the Korteweg–de Vries (KdV) equation and the Boussinesq equation were ob- tained. Following this work, Dai, Zhang, Fan, and Ma et al. extended the method to other equations such as Kadomtsev–Petviashvili equation, breaking soliton equation, Boussinesq equation, asymmetric Nizhnik–

Novikov–Veselov equation, and Bogoyavlenskii equa- tions [12–16]. The success of this method depends on circumventing the complicated algebro-geometric theory to directly give explicit quasi-periodic wave solutions. It can, however, be shown that all of the parameters appearing in the periodic wave solutions

are conditionally free variables. In the case of quasi- periodic solutions, it involves some Riemann constants which are difficult to be determined explicitly. To the knowledge of the authors, there is very few work available for constructing quasi-periodic solutions of differential-difference equations [16,17].

Based on the use of the Riemann theta function, we extend in this paper the Hirota bilinear method to construct quasi-periodic solutions of differential- difference equations. For illustration, a new inte- grable differential-difference equation [18] is chosen to demonstrate the feasibility of the proposed construc- tion method. It will be shown that the quasi-periodic wave solutions converge to the soliton solutions under certain conditions.

This paper is organized as follows. In Section2, we briefly introduce the bilinear form of differential- difference equation and the Riemann theta function.

The Hirota bilinear method and Riemann theta func- tion are then used in Section3to construct the quasi- periodic wave solutions for the differential-difference equation. Finally, we give an analysis on the asymp- totic behaviour of the quasi-periodic wave solutions in the last Section4in which it has rigorously been shown that the periodic solutions tend to the well-known soli- ton solutions under a ‘small amplitude’ limit.

c

2012 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen·http://znaturforsch.com

2. Bilinear Form and Riemann Theta Function We first consider an integrable differential- difference equation [18] whose bilinear form is

DtDx+ADtsinh(D_n)

−4 sinh² 1

2D_n

+c

f(n)·f(n) =0, (1)

whereAis an arbitrary constant andcis an integration constant. The bilinear differential operatorD_x,D_t, and the difference operator e^Dn are defined respectively by D^m_xD_t^m0f(x,t)·g(x,t)

= (∂_x−∂_x⁰)^m(∂_t−∂_t⁰)^m0f(x,t)g(x⁰,t⁰)|_x⁰_=x,t⁰_=t, e^δDnf(n)·g(n) =e^{δ(∂n−∂n0)}f(n)g(n⁰)|_n0=n

= f(n+δ)g(n−δ), sinh(δD_n)f(n)·g(n) =1

2(e^δDn−e^−δDn)f(n)·g(n)

=1

2(f(n+δ)g(n−δ)−f(n−δ)g(n+δ)).

(2)

The bilinear form (1) arises from many famous differential-difference equations. In particular, if f(n;x,t) = f(n;t), (1) becomes a special case of an ex- tended Lotka-Volterra equation [19]; whenA=0 and taking the transformation of the solution

τn= f(n−1)f(n+1)

f²(n) , (3)

(1) becomes the two-dimensional Toda equation [2]

∂²τn

∂x∂t =exp(τ_n−1−τ_n)−exp(τ_n−τn+1). (4) It had been shown in [18] that (1) is integrable in the sense of B¨acklund transformation.

The operatorsD_x,D_t, e^Dn, and sinh(δD_n)have the following nice properties when acting on exponential functions:

D^m_xD^m_t⁰e^ξ1·e^ξ2 = (α₁−α2)^m(ω₁−ω₂)^m0e^ξ1+ξ2, e^δDne^ξ1·e^ξ2=e^{δ(ν1−ν2)}e^ξ1+ξ2,

sinh(δD_n)e^ξ1·e^ξ2 =sinh[δ(ν₁−ν₂)]e^ξ1+ξ2, (5)

whereξj=αjx+ωjt+νjn+σj,j=1,2. More gen- erally, we have

G(D_x,Dt,sinh(δDn))e^ξ1·e^ξ2

=G(α₁−α₂,ω₁−ω₂,sinh(δ(ν₁−ν₂)))e^ξ1+ξ2, (6)

whereG(D_x,D_t,sinh(δDn))is a polynomial function with respect to the operatorsD_x,D_t, and sinh(δD_n).

In the special case ofc=0, (1) admits the following one-soliton solution:

f(n) =1+exp αx+4 sinh^{2 1}₂ν

α+Asinh(ν)t+νn+σ

!

. (7)

The following one-dimensional Riemann theta function plays a central role in the quasi-periodicity of the solutions:

ϑ(ξ,ε,s|τ) =

∑

m∈Z

exp[2πi(ξ+ε)(m+s)

−π τ(m+s)²],

(8) wherem∈Z,s,ε∈C, andξ =αx+ωt+νn+σ is a complex phase variable depending on the continuous variablesx,t, and discretized variablen. Here,τ>0 is called the period matrix of the Riemann theta function.

For simplicity, in the case whens=ε=0, we denote ϑ(ξ,τ) =ϑ(ξ,0,0|τ). (9) Definition 1. A functiong(t)onCis said to be quasi- periodic intwith fundamental periodsT1, . . . ,T_k∈Cif T₁, . . . ,T_kare linearly dependent overZand there exists a functionG(y₁, . . . ,y_k)∈C^k, such that

G(y₁, . . . ,y_j+T_j, . . . ,y_k)

=G(y₁, . . . ,y_j, . . . ,y_k), (10)

for all(y₁, . . . ,y_k)∈C^k. If we denote

G(t, . . . ,t, . . . ,t) =g(t), (11) theng(t) becomes periodic with the periodT if and only ifT_j=m_jT for somem_j∈Z.

Proposition 1. The Riemann theta functionϑ(ξ,τ) defined in (8) has the periodic properties [21,22]

ϑ(ξ+1+iτ,τ)

=exp(−2πiξ+π τ)ϑ(ξ,τ). (12) Proposition 2. The meromorphic functions F(ξ)on Csatisfy:

(i) F(ξ) =∂²

ξlnϑ(ξ,τ), ξ ∈C; (ii) F(ξ) =∂_ξlnϑ(ξ+e,τ)

ϑ(ξ+h,τ), ξ,e,h∈C; (iii) F(ξ) =ϑ(ξ+e,τ)ϑ(ξ−e,τ)

ϑ(ξ,τ)² , ξ,e∈C, (13)

which implies that

F(ξ+1+iτ) =F(ξ), ξ ∈C. (14) In other words, the meromorphic functions F(ξ)are quasi-periodic functions with two fundamental periods 1 and iτ[17].

3. Quasi-Periodic Solution

Consider the Riemann theta function solution for the differential-difference equation in bilinear form (1),

f(n) =ϑ(ξ,τ) =

∞ m=−∞

∑

e^{2πimξ−πm2τ}, wherem∈Z,τ>0, andξ =αx+ωt+νn+σ.

Substitute the abovef(n)into (1) gives

G(D_x,Dt,sinh(δDn))f(n)·f(n) =G(D_x,Dt,sinh(δDn))

∞

∑

m⁰=−∞

e^{2πim0ξ−πm02τ}·

∞ m=−∞

∑

e^{2πimξ−πm2τ}

=

∞

∑

m⁰=−∞

∞ m=−∞

∑

G(D_x,D_t,sinh(δD_n))e^{2πim0ξ−πm02τ}·e^{2πimξ−πm2τ}

=

∞

∑

m⁰=−∞

∞ m=−∞

∑

G 2πi(m⁰−m)α,2πi(m⁰−m)ω,sinh[2πiδ ν(m⁰−m)]

e^{2πi(m0+m)ξ−π(m02+m2)τ}

m⁰+m=l⁰

=

∞

∑

l⁰=−∞

∞

∑

m⁰=−∞

G(2πi(2m⁰−l⁰)α,2πi(2m⁰−l⁰)ω,sinh[2πiδ ν(2m⁰−l⁰)])e^{2πil0ξ−π[m02+(l0−m0)2]τ}

l⁰=2l+µ

=

∞ l=−∞

∑ ∑

µ=0,1

∞

∑

m⁰=−∞

G 4πi

m⁰−l−µ 2

α,4πi

m⁰−l−µ 2

ω,sinhh

4πiδ ν

m⁰−l−µ 2

i

·e^4πi(l+^µ₂)ξ−π[m⁰²+(2l+µ−m⁰)²]τ.

(15)

Letm⁰=h+l, and using the relations h+l=h

h−µ 2 i

+h l+µ

2 i

, h−l−µ=h

h−µ 2

i−h l+µ

2 i

, (16)

we finally obtain that

G(D_x,D_t,sinh(δD_n))f(n)·f(n)

=

∞ l=−∞

∑

(

∑

µ=0,1

∞ h=−∞

∑

G 4πi

h−µ 2

α,4πi h−µ

2

ω,sinhh

4πiδ ν h−µ

2 i

e^−2π(^h−µ₂)²τ

)

·e^4πi(l+^µ₂)ξ−2π(l+^µ₂)²τ

=

∞ l=−∞

∑

C(α,ω,ν,µ)e^4πi(^l+µ₂)ξ−2π(^l+µ₂)²τ,

(17)

where

C(α,ω,ν,µ) =

∑

µ=0,1

∞ h=−∞

∑

G 4πi

h−µ 2

α,4πi h−µ

2

ω,sinhh

4πiδ ν h−µ

2 i

e^−2π(^h−µ₂)²τ. (18)

It can be observed that if the following equa- tions C(α,ω,ν,µ) =0 are satisfied, for all possible combinationsµ=0,1, thenϑ(ξ,τ)is a solution of the

bilinear equation (1). On the other hand, the equa- tionsG(α,ω,ν,0) =0 andG(α,ω,ν,1) =0 can be explicitly written as

∞ h=−∞

∑

{−16π²h²ω α+4Aπihωsinh(4πiνh)−4sinh²(2πiνh) +c}e^−2πh2τ=0, (19)

∞ h=−∞

∑

(

−16π²

h−1 2

2

ω α+4Aπi

h−1 2

ωsinh

4πiν

h−1

2

−4sinh²

2πiν

h−1 2

+c

)

·e^−2π(^h−1₂)²τ=0, (20)

i.e.

ϑ₁⁰⁰(0,λ)ω α+Asinh(D_n)ϑ₁⁰(0,λ)ω

−4 sinh² 1

2D_n

ϑ1(0,λ) +cϑ₁(0,λ) =0, (21)

ϑ₂⁰⁰(0,λ)ω α+Asinh(D_n)ϑ₂⁰(0,λ)ω

−4 sinh² 1

2Dn

ϑ2(0,λ) +cϑ2(0,λ) =0, (22)

where the prime denotes the partial derivate∂_ξ and λ=e^{−12π τ},

ϑ₁(ξ,λ) =ϑ(2ξ,2τ)

=

+∞

h=−∞

∑

λ^4h

2exp(4πihξ),

ϑ₂(ξ,λ) =ϑ

2ξ,0,−1 2

2τ

=

+∞

h=−∞

∑

λ^(2h−1)

2exp[2πi(2h−1)ξ].

(23)

By introducing the notations

a₁₁=ϑ₁⁰⁰(0,λ)α+Asinh(D_n)ϑ₁⁰(0,λ), a₁₂=ϑ1(0,λ),

a₂₁=ϑ₂⁰⁰(0,λ)α+Asinh(D_n)ϑ₂⁰(0,λ), a₂₂=ϑ2(0,λ),

b1=4 sinh² 1

2Dn

ϑ1(0,λ), b₂=4 sinh²

1 2D_n

ϑ₂(0,λ),

(24)

the system (21) – (22) admits an explicit solution ω= b1a22−b2a12

a₁₁a₂₂−a₁₂a₂₁,

c= b2a11−b1a21

a₁₁a₂₂−a₁₂a₂₁.

(25)

Finally, we obtain the quasi-periodic solutions for the integral differential-difference lattice equation (1)

f(n) =

∞ m=−∞

∑

e^{2πimξ−πm2τ}, (26)

whereξ=αx+ωt+νn+σ,α,ν, andσare arbitrary constants,ωandcare given by (25).

4. Asymptotic Properties

In this section, an analysis on the asymptotic prop- erties of the one-periodic wave solution is given. It will be shown that the one-soliton solution can be obtained as a limiting case of the one-periodic wave solution (19). We will directly use the system (25) to analyse the asymptotic properties of the periodic solution, which is easier and more effective than our original method proposed in [14,15]. The relations between these two solutions are established as follows.

Theorem 1. Suppose that the vector(ω,c)is a so- lution of the system (25), and for the periodic wave solution (26), we let

α⁰=2πiα, ν⁰=2πiν, σ⁰=2πiσ−π τ, (27) whereα,ν, andδ are given in (26). Then we have the following asymptotic properties:

c→0, 2πiξ−π τ→η, η=α⁰x+ 4 sinh^{2 1}₂ν⁰

α⁰−Asinh(ν⁰)t+ν⁰n+σ⁰, θ(ξ,τ)→1+e^η as λ →0.

(28)

In other words, the periodic solution (26) tends to the soliton solution (7) under a small amplitude limit.

Proof. Since the coefficients of system (24) are power series about λ, its solution (ω,c) is also a series

aboutλ. The coefficients of the system (24) can then be explicitly expanded as follows:

a₁₁= [−32π²α+4Aπi sinh(4πiν)]λ⁴+· · ·, a12=1+2λ⁴+2λ¹⁶· · ·,

a₂₁= [−8π²α+4Aπi sinh(2πiν)]λ+· · ·, a22=2λ+2λ³+2λ⁹· · ·,

b₁=8 sinh⁴(2πiν)λ⁴+· · ·, b₂=8 sinh⁴(πiν)λ+· · ·.

(29)

-10 -5 0

5

10

x -10

0 10 t 0.99

1 1.01 τn

10 -5 0

5 x

-20 -10 0 10 20

-20 -10 0 10 20

-20 -10 10 20 t

0.985 0.99 0.995 1.005 1.01 1.015 τn

-20 -10 10 20 x

0.985 0.99 0.995 1.005 1.01 1.015 τn

(a) (b)

(c) (d)

Fig. 1 (colour online). Quasi-periodic wave for the two-dimensional Toda equation (4): (a) perspective view of wave, (b) overhead view of wave, with contour plot shown, (c) alongt-axis, (d) alongx-axis, whereα=0.1,ν=0.1,n=10,σ=0, andτ=1.

Let the solution of the system (25) be in the form ω=ω0+ω1λ+ω₂λ²+· · ·=ω0+o(λ), c=c₀+c₁λ+c₂λ²+· · ·=c₀+o(λ). (30) Substituting the expansions (29) and (30) into the sys- tem (25) and lettingλ→0, we immediately obtain the following relations:

c₀=0, w₀= 8 sinh⁴(πiν)

−8π²α+4Aπi sinh(2πiν). (31)

-50 -25

0 25

50 x

-50

-250 25 50 t 1.0011.002τn1

0 -25

0 25

50 x

-60 -40 -20 20 40 60 t 1.0005

1.001 1.0015 1.002 1.0025 τn

-60 -40 -20 20 40 60 x 1.0005

1.001 1.0015 1.002 1.0025 τn

(a)

(b) (c)

Fig. 2 (colour online). Solitary wave for the two-dimensional Toda equation (4): (a) perspective view of wave, (b) alongt-axis, (c) alongx-axis, whereα=0.1,ν=0.1,n=10,σ=0, andτ=1.

Combining (30) and (31) leads to 2πiw→4 sinh^{2 1}₂ν⁰

α⁰+Asinh(ν⁰), c→0 as λ→0,

(32)

or equivalently,

ξˆ=2πiξ−π τ (33)

=α⁰x+4 sinh^{2 1}₂ν⁰

α⁰+Asinh(ν⁰)t+ν⁰n+σ⁰→η.

It remains to consider the asymptotic properties of the one-periodic wave solution (26) under the limitλ→0.

For this purpose, we expand the Riemann theta func- tion ϑ(ξ,τ) and make use of the expression (33) to

obtain

ϑ(ξ,τ) =1+λ²

e^2πiξ+e^−2πiξ +λ⁸

e^4πiξ+e^−4πiξ +· · ·

=1+e^ξˆ+λ⁴

e^−ξˆ+e^{2 ˆξ} +· · ·

→1+e^η as λ→0,

(34)

which complete the proof for the theorem. We con- clude that the periodic solution (26) tends to the soliton solution (7) as the amplitudeλ→0.

The bilinear form (1) in general arises from many famous differential-difference equations. For example, whenA=0 and taking the transformation of the so- lution (3), the bilinear form (1) becomes the two-

dimensional Toda equation (4). From Theorem1, we can directly obtain the quasi-periodic solution τn of the two-dimensional Toda equation (4). The quasi- periodic and the corresponding soliton solutions of the two-dimensional Toda equation have been presented in Figures1and2.

5. Conclusion

In this paper, we give a construction method for ob- taining quasi-periodic wave solutions of differential- difference equations. Similarly, multi-periodic wave solutions of differential-difference equations can be constructed by using the following multi-dimensional

Riemann theta function:

ϑ(ξξξ,τ) =

∑

m m m∈Z^N

exp{2πihξξξ,mmmi −πhτττmmm,mmmi}, (35) whereξξξ = (ξ₁,ξ2, . . . ,ξN)^T∈C^N,mmm= (m₁,m₂, . . . , m_N)^T∈Z^N,ξ_j=α_jx+ω_jt+ν_jn+σ_j, j=1, . . . ,N, τ is aN×N symmetric positive definite matrix. The inner product is defined by

hfff,gggi=f₁g₁+f₂g₂+· · ·+f_Ng_N, (36) for two vectors fff = (f₁,f₂, . . . ,f_N)^T andggg= (g₁,g₂, . . . ,g_N)^T.

In order that the multi-dimensional Riemann theta function (35) satisfy the bilinear equation (1), from (18) we have

∑

µ=0,1

∞

∑

h₁,...,h_N=−∞

G 4πi

N

∑

j=1

h_j−µ_j

2

αj,4πi

N

∑

j=1

h_j−µ_j

2

ωj,sinh

"

4πiδ

N

∑

j=1

νj

h_j−µ_j

2

#!

·exp

"

−2π

N

∑

j,l=1

h_j−µ_j

2

τjk

h_k−µ_k

2

#

=0.

(37)

Obviously, in the case of differential-difference equa- tions, the number of constraint equations of the type (17) is 2^N. On the other hand, we have parameters τ_jk =τ_{k j}, αj, ωj, νj, and c whose total number is

1

2N(N+1) +3N+1. Among which 3Nparametersτj j, αj, andνjare taken to be the given parameters related to the amplitudes and wave numbers (or frequencies) of N-periodic waves, and ¹₂N(N−1) parametersτjk

implicitly appear in series form, which in general can- not be solved explicitly. Hence, the number of the ex- plicit unknown parameters is onlyN+1. The number of equations is larger than the number of unknown pa-

rameters in the case whenN≥2. In this paper, we con- sider the one-periodic wave solution of (1), which be- longs to the case whenN=1. The case whenN≥2 will be considered in our future work.

Acknowledgements

The work described in this paper was partially sup- ported by a grant from CityU (Project No. 7002564).

The work was also partially supported by the Leading Academic Discipline Program, 211 Project for Shang- hai University of Finance and Economics.

[1] R. Hirota and J. Satsuma, Prog. Theor. Phys. 57, 797 (1977).

[2] R. Hirota, The Direct Methods in Soliton Theory, Springer-Verlag, Berlin 2004.

[3] X. B. Hu and P. A. Clarkson, J. Phys. A: Math. Theor.

28, 5009 (1995).

[4] X. B. Hu, C. X. Li, J. J. C. Nimmo, and G. F. Yu, J. Phys. A: Math. Theor.38, 195 (2005).

[5] R. Hirota and Y. Ohta, J. Phys. Soc. Jpn.60, 798 (1991).

[6] D. J. Zhang, J. Phys. Soc. Jpn.71, 2649 (2002).

[7] W. X. Ma, Mod. Phys. Lett. B19, 1815 (2008).

[8] K. Sawada and T. Kotera, Prog. Theor. Phys.51, 1355 (1974).

[9] Y. Ohta, K. I. Maruno, and B. F. Feng, J. Phys. A: Math.

Theor.41, 355205 (2008).

[10] A. Nakamura, J. Phys. Soc. Jpn.47, 1701 (1979).

[11] A. Nakamura, J. Phys. Soc. Jpn.48, 1365 (1980).

[12] H. H. Dai, E. G. Fan, and X. G. Geng, arxiv.org/pdf/

nlin/0602015.

[13] Y. Zhang, L. Y. Ye, Y. N. Lv, and H. Q. Zhao, J. Phys.

A: Math. Theor.40, 5539 (2007).

[14] E. G. Fan, J. Phys. A: Math. Theor.42, 095206 (2009).

[15] W. X. Ma, R. G. Zhou, and L. Gao, Mod. Phys. Lett. A 24, 1677 (2009).

[16] E. G. Fan and Y. C. Hon, Phys. Rev. E 78, 036607 (2008).

[17] Y. C. Hon and E. G. Fan, Z. Y. Qin, Mod. Phys. Lett. B 22, 547 (2008).

[18] E. G. Fan and K. W. Chow, Phys. Lett. A 374, 3629 (2010).

[19] X. B. Hu, P. A. Clarkson, and R. Bulloughx, J. Phys. A:

Math. Theor.30, L669 (1997).

[20] K. Narita, J. Phys. Soc. Jpn.51, 1682 (1982).

[21] H. M. Farkas and I. Kra, Riemann Surfaces, Springer- Verlag, New York 1992.

[22] D. Mumford, Theta Lectures on Theta II, Birkh¨auser, Boston 1983.