Bäcklund Transformation and Quasi-Periodic Solutions for a Variable-Coefficient Integrable Equation

Abstract

Binary Bell polynomials are applied to construct bilinear formalism, bilinear Bäcklund transformation, Lax pair, and infinite conservation laws of the generalized variable-coefficient fifth-order Korteweg-de Vries equation. In the meantime, quasi-periodic wave solutions for the equation are obtained by using the Riemann theta function. The asymptotic properties of one-periodic wave solution and two-periodic wave solutions are also established, respectively.

1. Introduction

Nonlinear evolution equations (NLEEs) have attracted intensive attention in the past few decades, since they occur in a variety of physical applications. It is always important to search for explicit and exact solutions. Various kinds of exact solutions such as soliton, peakon, complexiton, rational, periodic, and quasi-periodic solutions have been presented for NLEEs. Successful methods include the inverse scattering method [1], the Darboux transformation [2–4] and the Bäcklund transformation [5, 6], the Hirota method [7, 8], and algebrogeometrical approach [9–11]. Among the abovementioned methods, the Hirota method is a powerful approach to construct exact solution of nonlinear equations. By applying the Hirota method, people obtained a series of multisoliton solutions and rational solutions of many nonlinear equations in a systematic way. Unfortunately, this method relies on particular skills, appropriate exchange formulas, and complex calculations. On the other hand, in recent years, Lambert, Gilson et al. proposed an alternative procedure based on the use of the Bell polynomials to obtain parameter families of bilinear Bäcklund transformation and Lax pairs for soliton equations in quick and short way [12–14]. Fan developed this method to find infinite conservation laws of soliton equations [15–17] and proposed the super Bell polynomials [18, 19]. Ma systematically analyzed the connection between Bell polynomials and new bilinear equations [20].

From bilinear forms, Nakamura proposed a convenient way to construct a kind of quasi-periodic solutions of nonlinear equation in his two serial papers [21, 22], where the quasi-periodic wave solutions of the KdV equation and the Boussinesq equation were obtained by using the Riemann theta function. Recently, Hon et al. have extended this method to investigate the discrete Toda lattice [23], (2 + 1)-dimensional Bogoyavlenskiis breaking soliton equation [24], and the asymmetrical Nizhnik-Novikov-Veselov equation [25]. Ma et al. constructed one-periodic and two-periodic wave solutions to a class of (2 + 1)-dimensional Hirota bilinear equations [26]. Zhang et al. applied this method to get periodic wave solutions of the variable-coefficient mKdV equation [27].

Due to the inhomogeneities of media and nonuniformities of boundaries in various real physical situations, the variable-coefficient NLEEs are considered to be more realistic than constant-coefficient equations in describing a large variety of real phenomena; for example, many physical and mechanical situations are governed by variable-coefficient KdV equation, for example, the nonlinear excitations of a Bose gas of impenetrable bosons with longitudinal confinement, the nonlinear waves in types of rods [28–30]. Obviously, equations with variable-coefficient are much more complicated than constant-coefficient forms, and much attention has been paid to this subject [31–35]. In this paper, we will focus our study on the generalized variable-coefficient fifth-order Korteweg-de Vries equation such as the one given below: 𝑢_t+𝑎(𝑡)𝑢𝑢_𝑥𝑥𝑥+𝑏(𝑡)𝑢_𝑥𝑢_𝑥𝑥+𝑐(𝑡)𝑢²𝑢_𝑥+𝑑(𝑡)𝑢𝑢_𝑥+𝑒(𝑡)𝑢_𝑥𝑥𝑥+𝑙(𝑡)𝑢_{𝑥𝑥𝑥𝑥𝑥}+𝑚(𝑡)𝑢+𝑛(𝑡)𝑢_𝑥=0, (1) where 𝑢 is a function of 𝑥 and 𝑡 and 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), 𝑑(𝑡), 𝑒(𝑡), 𝑙(𝑡), 𝑚(𝑡), and 𝑛(𝑡) are analytic functions of 𝑡. Since there are choices for the parameters, the variable-coefficient NLEEs can be considered as generalizations of the constant coefficient ones. Under certain constraint conditions, the variable-coefficient models may be proved to be integrable and given explicit analytic solutions [36]. The corresponding constraint conditions on (1) in this paper, which are obtained by the Painlev analysis [37] and conditions from the variable-coefficient models mapped to the completely integrable constant-coefficient counterparts [38], will be 𝑎(𝑡)(𝑡)=(𝑡)=𝑏15𝑙𝜌𝑒^{∫𝑚(𝑡)𝑑𝑡}(𝑡)=(𝑡),𝑐45𝑙𝜌²𝑒^{∫2𝑚(𝑡)𝑑𝑡},𝑑(𝑡)=𝑒(𝑡)=0, (2) where 𝜌≠0 is an arbitrary constant. The main goal of this paper is twofold. First, we apply the binary Bell polynomials to construct bilinear formalism, bilinear Bäcklund transformation, Lax pairs, and infinite conservation laws of (1) under condition (2). Second, we obtain the periodic wave solutions of the equation by using the Riemann theta function and discussing their asymptotic properties.

The organization of this paper is as follows. In Section 2, we briefly present necessary notations on binary Bell polynomial that will be used in this paper. In Section 3, we get bilinear formalism, bilinear Bäcklund transformation, Lax pairs, and infinite conservation laws of the generalized variable-coefficient fifth-order Korteweg-de Vries equation by utilizing the binary Bell polynomials. In Section 4, we apply Hirota’s bilinear method to construct one- and two-periodic wave solutions (1), respectively. Further we use a limiting procedure to analyze asymptotic behavior of the periodic wave solutions in detail. Finally, some conclusions are given in Section 5.

2. Binary Bell Polynomials

To begin with, we will give some basic concepts and notations about the Bell polynomials. For details, please refer to [11–13].

Let 𝑓=𝑓(𝑥₁,𝑥₂,…,𝑥_𝑙) be a 𝐶^∞ function with multivariables; the following polynomials 𝑌_{𝑛1𝑥1,…,𝑛𝑙𝑥𝑙}𝑓=𝑌_{𝑛1,…,𝑛𝑙}𝑓_{𝑟1𝑥1,…,𝑟𝑙𝑥𝑙}=exp−𝑓𝜕^𝑛1_𝑥1⋯𝜕^𝑛𝑙_𝑥𝑙exp𝑓 (3) are called the multidimensional Bell polynomials, where 𝑓_{𝑟1𝑥1,…,𝑟𝑙𝑥𝑙}=𝜕^𝑟1_𝑥1⋯𝜕^𝑟𝑙_𝑥𝑙𝑓,𝑟₁=0,…,𝑛₁,𝑟_𝑙=0,…,𝑛_𝑙. (4) For convenience, we denote the multidimensional Bell polynomials by 𝑌-polynomials.

For example, for the simplest case 𝑓=𝑓(𝑥), the one-dimensional Bell polynomials are 𝑌₁=𝑓_𝑥,𝑌₂=𝑓_2𝑥+𝑓²_𝑥,𝑌₃=𝑓_3𝑥+3𝑓_𝑥𝑓_3𝑥+𝑓³_𝑥,…. (5) For 𝑓=𝑓(𝑥,𝑡), the two-dimensional Bell polynomials are 𝑌_𝑥𝑓(𝑥),𝑌=𝑓_2𝑥𝑓=𝑓_2𝑥+𝑓²_𝑥,𝑌_2𝑥𝑓=𝑓_3𝑥+3𝑓_𝑥𝑓_2𝑥+𝑓³_𝑥,𝑌_𝑥,𝑡𝑓=𝑓_𝑥,𝑡+𝑓_𝑥𝑓_𝑡,𝑌_2𝑥,𝑡𝑓=𝑓_2𝑥,𝑡+𝑓_2𝑥𝑓_𝑡+2𝑓_𝑥,𝑡𝑓_𝑥+𝑓²_𝑥𝑓_𝑡,…. (6)

Based on the above Bell polynomials, the multidimensional binary Bell polynomials (𝒴-polynomials) can be defined as follows: 𝒴_{𝑛1𝑥1,…,𝑛𝑙𝑥𝑙}(𝑣,𝑤)=𝑌_{𝑛1𝑥1,…,𝑛𝑙𝑥𝑙}𝑓|_{𝑓𝑟1𝑥1𝑙𝑥𝑙,…,𝑟}=𝑣_{𝑟1𝑥1,…,𝑟𝑙𝑥𝑙},𝑟₁+⋯+𝑟_𝑙𝑤isodd,_{𝑟1𝑥1,…,𝑟𝑙𝑥𝑙},𝑟₁+⋯+𝑟_𝑙iseven. (7) The 𝒴-polynomials inherit the easily recognizable partial structure of the Bell polynomials. The first few lowest order binary Bell polynomials are 𝑦_𝑥(𝑣)=𝑣_𝑥,𝑦_2𝑥(𝑣,𝑤)=𝑤_2𝑥+𝑣²_𝑥,𝑦_𝑥,𝑡(𝑣,𝑤)=𝑤_𝑥,𝑡+𝑣_𝑥𝑣_𝑡,𝑦_3𝑥(𝑣,𝑤)=𝑣_3𝑥+3𝑣_𝑥𝑤_2𝑥+𝑣³_𝑥,𝑦_4𝑥(𝑣,𝑤)=𝑤_4𝑥+3𝑤²_2𝑥+4𝑣_𝑥𝑣_3𝑥+6𝑣²_𝑥𝑤_2𝑥+𝑣⁴_𝑥,𝑦_5𝑥(𝑣,𝑤)=𝑣_5𝑥+5𝑣_𝑥𝑤_4𝑥+10𝑣_3𝑥𝑤_2𝑥+10𝑣²_𝑥𝑣_3𝑥+15𝑣_𝑥𝑤²_2𝑥+10𝑣³_𝑥𝑤_2𝑥+𝑣⁵_𝑥. (8)

Theorem 1 (see [11]). The link between binary Bell polynomials 𝒴_{𝑛1𝑥1,…,𝑛𝑙𝑥𝑙}(𝑣,𝑤) and the standard Hirota bilinear equation 𝐷^𝑛1_𝑥1⋯𝐷^𝑛𝑙_𝑥𝑙𝐹⋅𝐺 can be given by an identity 𝒴_{𝑛1𝑥1,…,𝑛𝑙𝑥𝑙}𝐹𝑣=ln𝐺=(),𝑤=ln𝐹𝐺𝐹𝐺⁻¹𝐷^𝑛1_𝑥1⋯𝐷^𝑛𝑙_𝑥𝑙𝐹⋅𝐺, (9) in which 𝑛₁+𝑛₂+⋯+𝑛_𝑙≥1 and operators 𝐷_𝑥1⋯𝐷_𝑥𝑙 are classical Hirotas bilinear operators defined by 𝐷^𝑛1_𝑥1⋯𝐷^𝑛𝑙_𝑥𝑙=𝜕𝐹⋅𝐺_𝑥1−𝜕_𝑥′1^𝑛1⋯𝜕_𝑥𝑙−𝜕_𝑥′𝑙^𝑛𝑙×𝐹𝑥₁,…,𝑥_𝑙×𝐺(𝑥^′₁,…,𝑥^′_𝑙)_{𝑥′1=𝑥1,…,𝑥′𝑙=𝑥𝑙}. (10)

In the particular case, when 𝐹=𝐺, formula (9) becomes 𝐺⁻²𝐷^𝑛1_𝑥1⋯𝐷^𝑛𝑙_𝑥𝑙𝐺⋅𝐺=𝒴_{𝑛1𝑥1,…,𝑛𝑙𝑥𝑙}=0,𝑞=2ln𝐺0,𝑛₁+⋯+𝑛_𝑙𝑃isodd,_{𝑛1𝑥1,…,𝑛𝑙𝑥𝑙}𝑞,𝑛₁+⋯+𝑛_𝑙iseven, (11)

in which the 𝑃-polynomials can be characterized by an equally recognizable even part partitional structure 𝑃_2𝑥𝑞=𝑞_2𝑥,𝑃_𝑥,𝑡𝑞=𝑞_𝑥𝑡,𝑃_4𝑥𝑞=𝑞_4𝑥+3𝑞²_2𝑥,𝑃_6𝑥𝑞=𝑞_6𝑥+15𝑞_2𝑥𝑞_4𝑥+15𝑞³_2𝑥,…. (12)

This formulae will be used to obtain the bilinear Bäcklund transformations of the NLEEs. It means that once an NLEE is written in a combination form of the 𝒴-polynomials, then it can be easily transformed into the corresponding bilinear Bäcklund transformation form.

Theorem 2 (see [11]). The binary Bell polynomials 𝒴_{𝑛1𝑥1,…,𝑛𝑙𝑥𝑙}(𝑣,𝑤) can be separated into 𝑃-polynomials and 𝑌-polynomials: (𝐹𝐺)⁻¹𝐷^𝑛1_𝑥1⋯𝐷^𝑛𝑙_𝑥𝑙𝐹⋅𝐺=𝒴_{𝑛1𝑥1,…,𝑛𝑙𝑥𝑙}(𝑣,𝑤)|_{𝑣=ln𝐹/𝐺,𝑤=ln𝐹𝐺}=𝒴_{𝑛1𝑥1,…,𝑛𝑙𝑥𝑙}𝑣,𝑣+q_{𝑣=ln𝐹/𝐺,𝑞=2ln𝐺}=_{𝑛1+⋯+𝑛𝑙=even}^𝑛1_𝑟1=0⋯^𝑛𝑙_𝑟𝑙^𝑙_=0𝑖=1𝑛_𝑖𝑟_𝑖𝑃_{𝑟1𝑥1,…,𝑟𝑙𝑥𝑙}𝑞×𝑌_{(𝑛1−𝑟1)𝑥1,…,(𝑛𝑙−𝑟𝑙)𝑥𝑙}(𝑣). (13) Under the Hopf-Cole transformation 𝑣=ln𝜓, that is, 𝜓=𝐹/𝐺, multidimensional binary Bell polynomials 𝑌_{𝑛1𝑥1,…,𝑛𝑙𝑥𝑙}(𝑣) can be linearized into the following form: 𝑌_{𝑛1𝑥1,…,𝑛𝑙𝑥𝑙𝑣=ln𝜓}=𝜓_{𝑛1𝑥1,…,𝑛𝑙𝑥𝑙}𝜓. (14)
Formulae (13) and (14) provide the shortest way to the associated Lax system of nonlinear equations.

3. Bilinear Representation, Bäcklund Transformation, and Conservation Laws of (1)

In this section, we will systematically investigate bilinear representation, Bäcklund transformation, Lax pair, and infinite conservation laws of (1) based on the Bell polynomials.

3.1. Bilinear Representation

In order to detect the existence of the bilinear representation, we introduce a potential field 𝑞 by setting 𝑢=𝑠𝑞_2𝑥, (15) with 𝑠=𝑠(𝑡) being a free function with respect to 𝑡, which will be chosen appropriately so that (1) is related to the 𝑃-polynomials. Then substituting (15) into (1) and integrating with respect to 𝑥 and noting condition (2) yield 𝐸𝑞=𝑠^′(𝑡)𝑠(𝑡)+𝑚(𝑡)𝑞_𝑥+𝑞_𝑥𝑡+𝑛(𝑡)𝑞_2𝑥+𝑎(𝑡)𝑠(𝑡)𝑞_2𝑥𝑞_4𝑥+𝑐(𝑡)𝑠²(𝑡)𝑞³_2𝑥3+𝑙(𝑡)𝑞_6𝑥=0. (16) Comparing the fourth and the sixth terms of the above equation with formula (12) implies that we should require 𝑠(𝑡)=𝜌𝑒^{−∫𝑚(𝑡)𝑑𝑡}. The resulting equation is then cast into a combination form of the 𝑃-polynomials: 𝐸𝑞=𝑞_𝑥𝑡+𝑛(𝑡)𝑞_2𝑥+𝑙(𝑡)𝑞_6𝑥+15𝑞_2𝑥𝑞_4𝑥+15𝑞³_2𝑥=𝑃_𝑥𝑡𝑞+𝑛(𝑡)𝑃_2𝑥𝑞+𝑙(𝑡)𝑃_6𝑥𝑞=0. (17) Making a change of the dependent variable 𝑞=2ln𝐹⟺𝑢=𝑠𝑞_2𝑥=2𝜌𝑒^{−∫𝑚(𝑡)𝑑𝑡}(ln𝐹)_𝑥𝑥 (18) and noting property (11), we can obtain the bilinear representation of (1) as 𝐷_𝑥𝐷_𝑡+𝑙(𝑡)𝐷⁶_𝑥+𝑛(𝑡)𝐷²_𝑥𝐹⋅𝐹=0. (19)

Following the Hirota bilinear theory, one-soliton solution for (1) in explicit forms can be given as 𝑢₁=2𝜌𝑒^{−∫𝑚(𝑡)𝑑𝑡}𝜕²𝜕𝑥²log1+𝑒^𝜉1, (20) and two-soliton solutions are denoted by 𝑢₂=2𝜌𝑒^{−∫𝑚(𝑡)𝑑𝑡}𝜕²𝜕𝑥²log1+𝑒^𝜉1+𝑒^𝜉2+𝑒^{𝜉1+𝜉2+𝐴12}, (21) with 𝜉_𝑗=𝑘_𝑗𝑥−𝑘⁵_𝑗𝑙(𝑡)𝑑𝑡−𝑘_𝑗𝑛(𝑡)𝑑𝑡+𝜉⁰_𝑗,𝑒^𝐴12=(𝑘₁−𝑘₂)²𝑘²₁−𝑘₁𝑘₂+𝑘²₂𝑘₁+𝑘₂²𝑘²₁+𝑘₁𝑘₂+𝑘²₂, (22)(23) where 𝑘_𝑗 and 𝜉⁰_𝑗,  𝑗=1,2, are arbitrary real constants.

3.2. Bäcklund Transformation and Lax Pair

In the following, we search for the bilinear Bäcklund transformation and the Lax pair of (1). Let 𝑞^′=2ln𝐹 and 𝑞=2ln𝐺 be two different solutions of (17), respectively; we have the two-field condition 𝐸𝑞^′−𝐸𝑞=𝑞^′−𝑞_𝑥𝑡+𝑛(𝑡)𝑞^′−𝑞_2𝑥(𝑡)×𝑞+𝑙_′−𝑞_6𝑥+15𝑞^′_2𝑥𝑞^′_4𝑥−𝑞_2𝑥𝑞_4𝑥+15𝑞^′3_2𝑥−𝑞³_2𝑥=0. (24) If set 𝑞𝑣=^′−𝑞2𝐹=ln𝐺,𝑞𝑤=^′+𝑞2=ln(𝐹𝐺), (25) then (24) can be rewritten as 𝐸𝑞^′−𝐸𝑞2=𝑣_𝑥𝑡+𝑛(𝑡)𝑣_2𝑥+𝑙(𝑡)𝑣_6𝑥+15𝑤_4𝑥𝑣_2𝑥+15𝑤_2𝑥𝑣_4𝑥+45𝑣_2𝑥𝑤²_2𝑥+15𝑣³_2𝑥=𝜕_𝑥𝒴_𝑡(𝑣)+𝑛(𝑡)𝒴_𝑥(𝑣)+𝑙(𝑡)𝒴_5𝑥(()𝑣,𝑤)+𝑅𝑣,𝑤=0, (26) with 𝑅(𝑣,𝑤)=𝑙(𝑡)10𝑣_2𝑥𝑤_4𝑥+5𝑣_4𝑥𝑤_2𝑥+30𝑣_2𝑥𝑤²_2𝑥+15𝑣³_2𝑥−5𝑣_𝑥𝑤_5𝑥−10𝑣_3𝑥𝑤_3𝑥−30𝑣_𝑥𝑤_2𝑥𝑤_3𝑥−30𝑣²_𝑥𝑣_𝑥𝑥𝑤_2𝑥−10𝑣³_𝑥𝑤_3𝑥−5𝑣⁴_𝑥𝑣_2𝑥−20𝑣_𝑥𝑣_2𝑥𝑣_3𝑥−10𝑣²_𝑥𝑣_4𝑥. (27) Taking 𝒴_3𝑥(𝑣,𝑤)=𝜆, (28)

where 𝜆 is an arbitrary parameter. Then from (28), 𝑅(𝑣,𝑤) can be rewritten in the form 𝑅()5𝑣,𝑤=−2𝑙(𝑡)𝜕_𝑥𝒴_5𝑥(𝑣,𝑤)+3𝜆𝒴_3𝑥(𝑣,𝑤). (29) Then from (26)–(29), we deduce a coupled system of 𝒴-polynomials: 𝒴_3𝑥()𝜕𝑣,𝑤=𝜆,_𝑥𝒴_𝑡(𝑣)+𝜕_𝑥𝑛(𝑡)𝒴_𝑥(𝑣)−32𝑙(𝑡)𝒴_5𝑥()−𝑣,𝑤152𝑙(𝑡)𝜆𝒴_2𝑥(𝑣,𝑤)=0. (30) By application of the identity (13), the system (30) immediately leads to the following bilinear Bäcklund transformation: 𝐷³_𝑥𝐷−𝜆𝐹⋅𝐺=0,_𝑡+𝑛(𝑡)𝐷_𝑥−32𝑙(𝑡)𝐷⁵_𝑥−152𝜆𝑙(𝑡)𝐷²_𝑥+𝛽×𝐹⋅𝐺=0, (31) where 𝛽 is an arbitrary parameter.

By using the Hopf-Cole transformation 𝑣=ln𝜓, it follows from formulas (13) and (14) that 𝒴_𝑥(𝑣)=𝜓_𝑥𝜓,𝒴_2𝑥(𝑣,𝑤)=𝑞_2𝑥+𝜓_2𝑥𝜓,𝒴_𝑡(𝑣)=𝜓_𝑡𝜓,𝒴_3𝑥(𝑣,𝑤)=3𝑞_2𝑥𝜓_𝑥𝜓+𝜓_3𝑥𝜓,𝒴_5𝑥(𝑣,𝑤)=5𝑞_4𝑥𝜓_𝑥𝜓+15𝑞²_2𝑥𝜓_𝑥𝜓+10𝑞_2𝑥𝜓_3𝑥𝜓+𝜓_5𝑥𝜓; (32) therefore, system (30) is linearized into the corresponding Lax representation 𝐿₁𝜓≡3𝑞_2𝑥𝜕_𝑥+𝜕³_𝑥𝜕𝜓=𝜆𝜓,_𝑡+𝐿₂𝜓≡𝜕_𝑡+𝑛(𝑡)−152𝑙(𝑡)𝑞_4𝑥𝜕_𝑥−152𝜆𝑙(𝑡)𝜕²_𝑥−15𝑙(𝑡)𝑞_2𝑥𝜕³_𝑥−32𝑙(𝑡)𝜕⁵_𝑥+−152𝜆𝑙(𝑡)𝑞_2𝑥+𝛽𝜓. (33) It is easy to check that the integrability condition 𝐿₁−𝜆,𝜕_𝑡+𝐿₂𝜓=0 (34) is satisfied if 𝑞_2𝑥=(𝑒^{∫𝑚(𝑡)𝑑𝑡}/𝜌)𝑢 and 𝑢 is a solution of the generalized variable-coefficient fifth-order Korteweg-de Vries (1).

3.3. Infinite Conservation Laws

Next, through the Bell-polynomial-type Bäcklund transformation, we will perform the procedure of deriving the infinite sequence of conservation laws of (1) in the following form: 𝐼_𝑛,𝑡+𝐹_𝑛,𝑥=0,𝑛=1,2,…. (35) Let 𝑞𝜂=^′𝑥−𝑞_𝑥2; (36) it follows from relation (25) that 𝑣_𝑥=𝜂,𝑤_𝑥=𝑞_𝑥+𝜂. (37) Rewrite (30) in the conserved form 𝒴_3𝑥()𝜕𝑣,𝑤=𝜆,_𝑡𝒴_𝑥(𝑣)+𝜕_𝑥𝑛(𝑡)𝒴_𝑥(𝑣)−32𝑙(𝑡)𝒴_5𝑥()−𝑣,𝑤152𝜆𝑙(𝑡)𝒴_2𝑥(𝑣,𝑤)=0. (38) Substituting (37) into (38), we can obtain 𝜂_2𝑥+3𝜂𝜂_𝑥+3𝑞_2𝑥𝜂+𝜂³=𝜆=𝜀³,𝜂_𝑡+𝜕_𝑥𝑛(𝑡)3𝜂−2𝑙(𝑡)15𝜆𝑞_2𝑥+15𝜆𝜂_𝑥+𝜂_4𝑥+5𝑞_4𝑥𝜂−15𝑞²_2𝑥𝜂+5𝜂𝜂_3𝑥+5𝜆𝜂²−30𝑞²_2𝑥𝜂𝜂_𝑥+10𝜂²𝜂_2𝑥−15𝜂𝜂²_𝑥+𝜂⁵=0, (39)(40) where we have used (39) to get (40).

To proceed, inserting the expansion 𝜂=𝜀+^∞_𝑛=1𝐼_𝑛𝑞,𝑞_𝑥,…𝜀^−𝑛 (41) into (39) and equating the coefficients for power of 𝜀, we then obtain the recursion relations for the conserved densities 𝐼_𝑛: 𝐼₁=−𝑞_2𝑥𝑒=−^{∫𝑚(𝑡)𝑑𝑡}𝜌𝐼𝑢,₂=𝑞_3𝑥=𝑒^{∫𝑚(𝑡)𝑑𝑡}𝜌𝑢_𝑥,𝐼₃2=−3𝑞_4𝑥2=−3𝑒^{∫𝑚(𝑡)𝑑𝑡}𝜌𝑢_2𝑥,𝐼₄=13𝑞_5𝑥=13𝑒^{∫𝑚(𝑡)𝑑𝑡}𝜌𝑢_3𝑥, (42) and the recursion relation is given as 𝐼_𝑛1=−3𝐼_{𝑛−2,𝑥𝑥}−𝐼_{𝑛−1,𝑥}−𝑢𝐼_𝑛−2−^𝑛−3_𝑖=1𝐼_𝑖𝐼_{𝑛−2−𝑖,𝑥}−^𝑛−2_𝑖=1𝐼_𝑖𝐼_{𝑛−1−𝑖}−13_{𝑖+𝑗+𝑘=𝑛−2}𝐼_𝑖𝐼_𝑗𝐼_𝑘,𝑛=5,6,7,…. (43)

In addition, substituting (41) into (40) yields 𝐹₁=𝑛(𝑡)𝐼₁−32𝑙(𝑡)×15𝐼_4,𝑥+𝐼_1,4𝑥𝑒+5^{∫𝑚(𝑡)𝑑𝑡}𝜌𝑢_2𝑥𝐼₁𝑒−15^{2∫𝑚(𝑡)𝑑𝑡}𝜌²𝑢²𝐼₁+5𝐼_2,3𝑥+52𝐼₁𝐼₃+𝐼²₂+2𝐼₅𝑒−30^{2∫𝑚(𝑡)𝑑𝑡}𝜌²𝑢²𝐼_2,𝑥+102𝐼₁𝐼_1,2𝑥+𝐼_3,2𝑥−15𝐼²_1,𝑥+10𝐼³₁+15𝐼²₂+5𝐼₅,𝐹₂=𝑛(𝑡)𝐼₂−32𝑙(𝑡)×15𝐼_5,𝑥+𝐼_2,4𝑥𝑒+5^{∫𝑚(𝑡)𝑑𝑡}𝜌𝑢_2𝑥𝐼₂𝑒−15^{2∫𝑚(𝑡)𝑑𝑡}𝜌²𝑢²𝐼₂+5𝐼₁𝐼_1,3𝑥+𝐼_3,3𝑥+10𝐼₁𝐼₄+𝐼₂𝐼₃+𝐼₆𝑒−30^{2∫𝑚(𝑡)𝑑𝑡}𝜌²×𝑢²𝐼_3,𝑥+𝐼₁𝐼_1,𝑥+102𝐼₁𝐼_2,2𝑥+𝐼_4,2𝑥−30𝐼_1,𝑥𝐼_2,𝑥+30𝐼²₁𝐼₂+20𝐼₂𝐼₃+20𝐼₁𝐼₄+5𝐼₆,𝐹₃=𝑛(𝑡)𝐼₃−32𝑙(𝑡)×15𝐼_6,𝑥+𝐼_3,4𝑥𝑒+5^{∫𝑚(𝑡)𝑑𝑡}𝜌𝑢_2𝑥𝐼₃𝑒−15^{2∫𝑚(𝑡)𝑑𝑡}𝜌²𝑢²𝐼₃+5𝐼₁𝐼_2,3𝑥+𝐼₂𝐼_1,3𝑥+𝐼_4,3𝑥+52𝐼₁𝐼₅+2𝐼₂𝐼₄+𝐼²₃+2𝐼₇𝑒−30^{2∫𝑚(𝑡)𝑑𝑡}𝜌²𝑢²𝐼_4,𝑥+𝐼₁𝐼_2,𝑥+𝐼₂𝐼_1,𝑥+102𝐼₁𝐼_3,2𝑥+2𝐼₃𝐼_1,2𝑥+𝐼²₁𝐼_1,2𝑥+𝐼_5,2𝑥−15𝐼²_2,𝑥+𝐼₁𝐼²_2,𝑥+5𝐼⁴₁+30𝐼²₁𝐼₃+30𝐼₁𝐼²₂+10𝐼²₃+20𝐼₂𝐼₄+20𝐼₁𝐼₅+5𝐼₇,𝐹₄=𝑛(𝑡)𝐼₄−32𝑙(𝑡)×15𝐼_7,𝑥+𝐼_4,4𝑥𝑒+5^{∫𝑚(𝑡)𝑑𝑡}𝜌𝑢_2𝑥𝐼₄𝑒−15^{2∫𝑚(𝑡)𝑑𝑡}𝜌²𝑢²𝐼₄+5𝐼₁𝐼_3,3𝑥+𝐼₂𝐼_2,3𝑥+𝐼₃𝐼_1,3𝑥+𝐼_5,3𝑥+102𝐼₁𝐼₆+𝐼₂𝐼₅+𝐼₃𝐼₄+𝐼₈𝑒−30^{2∫𝑚(𝑡)𝑑𝑡}𝜌²𝑢²×𝐼_5,𝑥+𝐼₁𝐼_3,𝑥+𝐼₃𝐼_1,𝑥+𝐼₂𝐼_2,𝑥+102𝐼₁𝐼_4,2𝑥+2𝐼_1,2𝑥𝐼₁𝐼₂+𝐼₄+𝐼_2,2𝑥𝐼²₁+2𝐼₃+𝐼_6,2𝑥−152𝐼_2,𝑥𝐼_3,𝑥+2𝐼_1,𝑥𝐼²_4,𝑥+2𝐼₁𝐼_1,𝑥𝐼_2,𝑥+𝐼₂𝐼²_1,𝑥+20𝐼³₁𝐼₂+60𝐼₁𝐼₂𝐼₃+10𝐼³₂+30𝐼²₁𝐼₄+20𝐼₂𝐼₅+20𝐼₁𝐼₆+20𝐼₃𝐼₄+5𝐼₈,𝐹_𝑛=𝑛(𝑡)𝐼_𝑛−32𝑙(𝑡)×15𝐼_𝑛+3,𝑥+𝐼_𝑛,4𝑥𝑒+5^{∫𝑚(𝑡)𝑑𝑡}𝜌𝑢_2𝑥𝐼_𝑛𝑒−15^{2∫𝑚(𝑡)𝑑𝑡}𝜌²𝑢²𝐼_𝑛+5^𝑛−1_𝑘=1𝐼_𝑘𝐼_{𝑛−𝑘,3𝑥}+𝐼_𝑛+1,3𝑥+5^𝑛+2_𝑘=1𝐼_𝑘𝐼_{𝑛+3−𝑘}+2𝐼_𝑛+4𝑒−30^{2∫𝑚(𝑡)𝑑𝑡}𝜌²𝑢²×^𝑛−1_𝑘=1𝐼_𝑘𝐼_{𝑛−𝑘,𝑥}+𝐼_𝑛+1,𝑥+10×_{𝑖+𝑗+𝑘=𝑛}𝐼_𝑖𝐼_𝑗𝐼_𝑘,𝑥+2^𝑛_𝑘=1𝐼_𝑘𝐼_{𝑛+1−𝑘,𝑥}+𝐼_𝑛+2,𝑥−15_{𝑖+𝑗+𝑘=𝑛}𝐼_𝑖𝐼_𝑗,𝑥𝐼_𝑘,𝑥+^𝑛_𝑘=1𝐼_𝑘,𝑥𝐼_{𝑛+1−𝑘,𝑥}+_{𝑖+𝑗+𝑘+𝑙+𝑚=𝑛}𝐼_𝑖𝐼_𝑗𝐼_𝑘𝐼_𝑙𝐼_𝑚+_{𝑖+𝑗+𝑘+𝑙=𝑛+1}𝐼_𝑖𝐼_𝑗𝐼_𝑘𝐼_𝑙+_{𝑖+𝑗+𝑘=𝑛+2}𝐼_𝑖𝐼_𝑗𝐼_𝑘+_{𝑖+𝑗=𝑛+3}𝐼_𝑖𝐼_𝑗+𝐼_𝑛+4,𝑛=4,5,6,…. (44) With the recursion formulae of 𝐼_𝑛 and 𝐹_𝑛 presented previously, the infinite conservation laws for (1) can be constructed.

4. Quasi-Periodic Wave Solutions and Asymptotic Properties

The quasi-periodic wave solutions of (1) are based on the following multidimensional Riemann theta function of genus 𝑁: 𝜗(𝜉)=𝜗(𝜉,𝜏)=_{𝑛∈𝑍𝑁}𝑒^{−𝜋⟨𝜏𝑛,𝑛⟩+2𝜋𝑖⟨𝜉,𝑛⟩}. (45) Here the integer value vector 𝑛=(𝑛₁,…,𝑛_𝑁)^𝑇∈𝑍^𝑁, and complex phase variables 𝜉=(𝜉₁,…,𝜉_𝑁)^𝑇∈𝐶^𝑁. Moreover, for two vectors 𝑓=(𝑓₁,…,𝑓_𝑁)^𝑇 and 𝑔=(𝑔₁,…,𝑔_𝑁)^𝑇, their inner product is defined by 𝑓,𝑔=𝑓₁𝑔₁+𝑓₂𝑔₂+⋯+𝑓_𝑁𝑔_𝑁. (46) The 𝜏=(𝜏_𝑖𝑗) is a positive definite and real-valued symmetric 𝑁×𝑁 matrix, which we call the period matrix of the theta function. The entries 𝜏_𝑖𝑗 of the period matrix 𝜏 can be considered as free parameters of the theta function (45).

Now, we consider the solution for (1) in the following bilinear form: 𝐺𝐷_𝑥,𝐷_𝑡=𝐷_𝑥𝐷_𝑡+𝑙(𝑡)𝐷⁶_𝑥+𝑛(𝑡)𝐷²_𝑥+𝑐𝑓⋅𝑓=0, (47) where 𝑐 is the constant of integration.

4.1. Construction of One-Periodic Waves

In this section, we consider the one-periodic wave solutions for (1). When 𝑁=1, the theta function reduces the following Fourier series in 𝑛: 𝜗(𝜉,𝜏)=^∞_𝑛=−∞𝑒^{2𝜋𝑖𝑛𝜉−𝜋𝑛2𝜏}, (48) where the phase variable 𝜉=𝑘𝑥+∫𝜔𝑑𝑡+𝜉⁽⁰⁾ and the parameter 𝜏>0.

Substituting (48) into (47), we obtain 𝐺𝐷_𝑥,𝐷_𝑡𝜗()()=𝜉,𝜏⋅𝜗𝜉,𝜏_{∞∞𝑛=−∞𝑚=−∞}𝐺𝐷_𝑥,𝐷_𝑡𝑒^{2𝜋𝑖𝑛𝜉−𝜋𝑛2𝜏}𝑒^{2𝜋𝑖𝑚𝜉−𝜋𝑚2𝜏}=^∞∞_{𝑛=−∞𝑚=−∞}𝐺[2𝜋𝑖(𝑛−𝑚)𝑘,2𝜋𝑖(𝑛−𝑚)𝜔]×𝑒^{2𝜋(𝑛+𝑚)𝜉−𝜋(𝑛2+𝑚2)𝜏}_{𝑚=𝑚′−𝑛}=^∞_{𝑚′=−∞}^∞_𝑛=−∞𝐺2𝜋𝑖2𝑛−𝑚^′𝑘,2𝜋𝑖2𝑛−𝑚^′𝜔×𝑒^{−𝜋[𝑛2+(𝑛−𝑚′)2]𝜏}𝑒^{2𝜋𝑖𝑚′𝜉}=^∞_{𝑚′=−∞}𝐺𝑚^′𝑒^{2𝜋𝑖𝑚′𝜉}, (49) where 𝐺𝑚^′=^∞_𝑛=−∞𝐺2𝜋𝑖2𝑛−𝑚^′𝑘,2𝜋𝑖2𝑛−𝑚^′𝜔×𝑒^{−𝜋[𝑛2+(𝑛−𝑚′)2]𝜏}. (50) By shifting sum index as 𝑛=𝑛^′+1, we conclude that 𝐺𝑚^′=^∞_{𝑛′=−∞}𝐺2𝜋𝑖2𝑛^′−𝑚^′−2𝑘,2𝜋𝑖2𝑛^′−𝑚^′−2𝜔×𝑒^{−𝜋{𝑛′2+[𝑛′−(𝑚′−2)]2}𝜏}𝑒^{[−2𝜋(𝑚′−1)𝜏]}=𝐺𝑚^′−2𝑒^{−2𝜋(𝑚′−1)𝜏}==⋯𝐺(0)𝑒^{−𝜋𝑚′2(𝜏/2)},𝑚^′iseven,𝐺(1)𝑒^{−𝜋(𝑚′2−1)(𝜏/2)},𝑚^′isodd, (51) which imply that if 𝐺(0)=𝐺(1)=0, then it follows that 𝐺𝑚^′=0,𝑚^′∈𝑍, (52) and thus the theta function (48) is the exact solution of (47). In this way, we may let 𝐺(0)=^∞_𝑛=−∞−16𝜋²𝑛²𝑘𝜔−4096𝜋⁶𝑛⁶𝑘⁶𝑙(𝑡)−16𝜋²𝑛²𝑘²𝑛(𝑡)+𝑐𝑒^{−2𝜋𝑛2𝜏}=0,𝐺(1)=^∞_𝑛=−∞−4𝜋²(2𝑛−1)²𝑘𝜔−64𝜋⁶(2𝑛−1)⁶𝑘⁶𝑙(𝑡)−4𝜋²(2𝑛−1)²𝑘²𝑛(𝑡)+𝑐𝑒^{−𝜋(2𝑛2−2𝑛+1)𝜏}=0. (53) Denote 𝜆=𝑒^−𝜋𝜏,𝑎₁₁=−^∞_𝑛=−∞16𝜋²𝑛²𝑘𝜆^2𝑛2,𝑎₁₂=^∞_𝑛=−∞𝜆^2𝑛2,𝑎₂₂=^∞_𝑛=−∞𝜆^{2𝑛2−2𝑛+1},𝑎₂₁=−^∞_𝑛=−∞4𝜋²(2𝑛−1)²𝑘𝜆^{2𝑛2−2𝑛+1},𝑏₁=^∞_𝑛=−∞4096𝜋⁶𝑛⁶𝑘⁶𝑙(𝑡)+16𝜋²𝑛²𝑘²𝑛(𝑡)𝜆^2𝑛2,𝑏₂=^∞_𝑛=−∞64𝜋⁶(2𝑛−1)⁶𝑘⁶𝑙(𝑡)+4𝜋²(2𝑛−1)²×𝑘²𝑛(𝑡)𝜆^{2𝑛2−2𝑛+1}. (54) Then (53) can be written as 𝑎₁₁𝑎₁₂𝑎₂₁𝑎₂₂𝜔𝑐=𝑏₁𝑏₂. (55)

Notice that there are a lot of choices for the angular wave number 𝑘. The determinant of the coefficient matrix 𝐴(𝑘)=(𝑎_𝑖𝑗(𝑘))_2×2 is a polynomial in 𝑘; if det(𝐴(𝑘))≢0, then 𝐴₀∶={𝑘∈𝑅∣det(𝐴(𝑘))=0} (56) is either an empty set or a finite set, and so, there are real solutions (𝜔,𝑐) to the system (55) for 𝑘∉𝐴₀. Solving this system, we have 𝑏𝜔=¹𝑎₂₂−𝑏₂𝑎₁₂𝑎₁₁𝑎₂₂−𝑎₁₂𝑎₂₁. (57) Therefore we get a one-periodic wave solution of (1): 𝑢=2𝜌𝑒^{−∫𝑚(𝑡)𝑑𝑡}log𝜗(𝜉,𝜏)_𝑥𝑥, (58) where the parameter 𝜔 is given by (57).

4.2. Asymptotic Property of One-Periodic Waves

In the following, we further consider asymptotic properties of the one-periodic wave solution. It is shown that the soliton solution of (1) can be obtained as a limit of the one-periodic wave solution. The relation between these two solutions can be established as Theorem 3.

Theorem 3. Suppose that the vector (𝜔,𝑐)^𝑇 is a solution of the system (55), and for the quasi-periodic wave solution (58), we let 𝜉𝜉=^′+𝜏2𝜋𝑖2𝑖,𝜉^′=𝑘^′𝑥+𝜔^′𝑑𝑡+𝜉^(0)′, (59) where 𝑘^′=2𝜋𝑖𝑘,𝜔^′=2𝜋𝑖𝜔,𝜉^(0)′  and  =2𝜋𝑖𝜉⁽⁰⁾−𝜋𝜏. Then the one-periodic solution (58) tends to the one-soliton solution (20) under a small amplitude limit; that is, 𝑢⟶𝑢₁,𝑎𝑠𝜆⟶0. (60)

Proof. By using (54), we write functions 𝑎_𝑖𝑗,𝑏_𝑗,  𝑖,𝑗=1,2, as the series about 𝜆: 𝑎₁₁=−32𝜋²𝑘𝜆²+4𝜆⁸+9𝜆¹⁸,𝑎+⋯₁₂=1+2𝜆²+2𝜆⁸+2𝜆¹⁸+2𝜆³²𝑎+⋯,₂₁=−8𝜋²𝑘𝜆+9𝜆⁵+25𝜆¹³,𝑎+⋯₂₂=2𝜆+2𝜆⁵+2𝜆¹³+2𝜆²⁵𝑏+⋯,₁=8192𝜋⁶𝑘⁶𝑙(𝑡)+32𝜋²𝑘²𝑛(𝑡)𝜆²𝑏+⋯,₂=128𝜋⁶𝑘⁶𝑙(𝑡)+8𝜋²𝑘²𝑛(𝑡)𝜆+⋯. (61) Suppose that the solution of system (55) has the following form: 𝜔=𝜔₀+𝜔₁𝜆¹+𝜔₂𝜆²+⋯=𝜔₀+𝑜(𝜆), (62) substituting expansions (61) and (62) into system (55), and let 𝜆→0; we can obtain the following relation immediately: 𝜔₀=−16𝜋⁴𝑘⁵𝑙(𝑡)−𝑘𝑛(𝑡); (63) combining (59) and (63), we then obtain 𝜔^′=2𝜋𝑖𝜔⟶−32𝜋⁵𝑙(𝑡)𝑖𝑘⁵(𝑡)(𝑡)𝑘−2𝜋𝑛𝑖𝑘,=−𝑙^′5−𝑛(𝑡)𝑘^′. (64)

It remains to show that the one-periodic wave (58) degenerates to the one-soliton solution (20) under the limit 𝜆→0. We first expand the periodic function 𝜗(𝜉) in the form 𝜗(𝜉,𝜏)=1+𝜆𝑒^{2𝜋𝑖𝜉}+𝑒^{−2𝜋𝑖𝜉}+𝜆⁴𝑒^{4𝜋𝑖𝜉}+𝑒^{−4𝜋𝑖𝜉}+⋯. (65) By using transformation (59), it follows that 𝜗(𝜉,𝜏)=1+𝑒^𝜉′+𝜆²𝑒^−𝜉′+𝑒^2𝜉′+𝜆⁶𝑒^−2𝜉′+𝑒^3𝜉′+⋯,⟶1+𝑒^𝜉′,as𝜆⟶0. (66) Thus, we conclude that the one-periodic solution (58) may go to one-soliton solution (20) as the amplitude 𝜆→0.

4.3. Construction of Two-Periodic Waves

In the case when 𝑁=2, the Riemann theta function (45) takes the form 𝜗(𝜉,𝜏)=𝜗𝜉₁,𝜉₂,𝜏=_𝑛∈𝑍2𝑒^{−𝜋⟨𝜏𝑛,𝑛⟩+2𝜋𝑖⟨𝜉,𝑛⟩}, (67) where 𝑛=(𝑛₁,𝑛₂)^𝑇∈𝑍², 𝜉=(𝜉₁,𝜉₂)^𝑇∈𝐶², and 𝜉_𝑗=𝑘_𝑗𝑥+∫𝜔_𝑗𝑑𝑡+𝜉_𝑗⁽⁰⁾,  𝑗=1,2. 𝜏 is a positive definite and real-valued symmetric 2×2 matrix which can take the form 𝜏𝜏=¹¹𝜏₁₂𝜏₂₁𝜏₂₂,𝜏₁₁>0,𝜏₂₂>0,𝜏₁₁𝜏₂₂−𝜏²₁₂>0. (68)

Substituting (67) into (47), we have 𝐺𝐷_𝑥,𝐷_𝑡𝜗𝜉₁,𝜉₂,𝜏⋅𝜗𝜉₁,𝜉₂=,𝜏_{𝑚,𝑛∈𝑍2}𝐺(2𝜋𝑖⟨𝑛−𝑚,𝑘⟩,2𝜋𝑖⟨𝑛−𝑚,𝜔⟩)×𝑒^{2𝜋𝑖⟨𝜉,𝑛+𝑚⟩−𝜋(⟨𝜏𝑚,𝑚⟩+⟨𝜏𝑛,𝑛⟩)}_{𝑚=𝑚′−𝑛}=_{𝑚′∈𝑍2𝑛∈𝑍2}𝐺2𝜋𝑖2𝑛−𝑚^′,𝑘,2𝜋𝑖2𝑛−𝑚^′,𝜔×𝑒^{−𝜋[⟨𝜏(𝑛−𝑚′,𝑛−𝑚′⟩]+⟨𝜏𝑛,𝑛⟩]}⋅𝑒^{2𝜋𝑖⟨𝜉,𝑚′⟩}=_{𝑚′∈𝑍2}𝐺𝑚^′₁,𝑚^′₂𝑒^{2𝜋𝑖⟨𝑚′,𝜉⟩}, (69) where 𝐺(𝑚^′₁,𝑚^′₂) is the coefficient of 𝑒^{2𝜋𝑖⟨𝑚′,𝜉⟩}. For each fixed 𝑙=1,2, by shifting 𝑗th summation index as 𝑛_𝑗=𝑛^′_𝑗+𝛿_𝑗,𝑙 with 𝛿_𝑗,𝑙 representing Kronecker’s delta we obtain that 𝐺𝑚^′₁,𝑚^′₂=_𝑛∈𝑍2𝐺2𝜋𝑖2𝑛−𝑚^′,𝑘,2𝜋𝑖2𝑛−𝑚^′,𝜔×𝑒^{−𝜋[⟨𝜏(𝑛−𝑚′,𝑛−𝑚′⟩]+⟨𝜏𝑛,𝑛⟩]}=_{𝑛′∈𝑍2}𝐺2𝜋𝑖²_𝑗=12𝑛^′_𝑗−𝑚^′_𝑗−2𝛿_𝑗𝑙𝑘_𝑗,2𝜋𝑖²_𝑗=12𝑛^′_𝑗−𝑚^′_𝑗−2𝛿_𝑖𝑗𝜔_𝑗×exp−𝜋²_𝑗,ℎ=1𝑛^′_𝑗+𝛿_𝑗𝑙𝜏_𝑗ℎ𝑛^′_ℎ+𝛿_ℎ𝑙−𝜋²_𝑗,ℎ=1𝑚^′_𝑗+2𝛿_𝑗𝑙−𝑛^′_𝑗+𝛿_𝑗𝑙×𝜏_𝑗ℎ𝑚^′_ℎ−2𝛿_ℎ𝑙−𝑛^′_ℎ+𝛿_ℎ𝑙=𝐺𝑚^′₁−2,𝑚^′₂𝑒^{−2𝜋(𝜏11𝑚′1+𝜏12𝑚′2)+2𝜋𝜏11},𝑙=1,𝐺𝑚^′₁,𝑚^′₂−2𝑒^{−2𝜋(𝜏12𝑚′1+𝜏22𝑚′2)+2𝜋𝜏22},𝑙=2, (70) which implies that if the following equations are satisfied 𝐺(0,0)=𝐺(0,1)=𝐺(1,0)=𝐺(1,1)=0 (71) then we have 𝐺(𝑚^′₁,𝑚^′₂)=0, for all 𝑚^′₁,𝑚^′₂∈𝑍, and thus the function (67) is an exact solution of (47). By introducing the notations as 𝐴=𝑎_𝑗𝑙4×3,𝑏=𝑏₁,𝑏₂,𝑏₃,𝑏₄^𝑇,𝑎_𝑗1=−4𝜋²_{𝑛1,𝑛2∈𝑍2}2𝑛−𝑠^𝑗,𝑘2𝑛₁−𝑠^𝑗₁𝜀_𝑗(𝑛),𝑎_𝑗2=−4𝜋²_{𝑛1,𝑛2∈𝑍2}2𝑛−𝑠^𝑗,𝑘2𝑛₂−𝑠^𝑗₂𝜀_𝑗(𝑛),𝑎_𝑗3=_{𝑛1,𝑛2∈𝑍2}𝜀_𝑗(𝑛),𝑏_𝑗=−𝑙(𝑡)(2𝜋𝑖)⁶2𝑛−𝑠^𝑗,𝑘⁶−𝑛(𝑡)(2𝜋𝑖)²×2𝑛−𝑠^𝑗,𝑘²,𝜀_𝑗(𝑛)=𝜆^{𝑛21+(𝑛1−𝑠𝑗1)2}₁𝜆^{𝑛22+(𝑛2−𝑠𝑗2)2}₂𝜆^{𝑛1𝑛2+(𝑛1−𝑠𝑗1)(𝑛2−𝑠𝑗2)}₁,𝜆₁=𝑒^{−𝜋𝜏11},𝜆₂=𝑒^{−𝜋𝜏22},𝜆₃=𝑒^{−2𝜋𝜏12},𝑠^𝑗=𝑠^𝑗₁,𝑠^𝑗₂𝑠,𝑗=1,2,3,4,₁=(0,0),𝑠²=(),𝑠1,0₃=(0,1),𝑠⁴=(1,1), (72)

(71) can be written as a linear system: 𝐴𝜔₁,𝜔₂,𝑐^𝑇=𝑏. (73) Notice that both 𝐴 and 𝑏 depend on 𝑘, so the solution of (73) also depends on 𝑘. And if rank(𝐴(𝑘))=rank(𝐴(𝑘),𝑏(𝑘))=3, (74) then there is a unique nonzero solution of (𝜔₁,𝜔₂,𝑐) to the system (73). Solving this system, we can get a two-periodic wave solution of (1): 𝑢=2𝜌𝑒^{−∫𝑚(𝑡)𝑑𝑡}log𝜗𝜉₁,𝜉₂,𝜏_𝑥𝑥, (75) with 𝜗(𝜉₁,𝜉₂) and 𝜔₁,𝜔₂,𝑐 given by (67) and (73), respectively, while 𝑘=(𝑘₁,𝑘₂) is determined by (74).

4.4. Asymptotic Property of Two-Periodic Waves

In this subsection, we consider the asymptotic properties of the two-periodic solution (75). In a similar way to Theorem 3, we can establish the relation between the two-periodic solution (75) and the two-soliton solution (21) as follows.

Theorem 4. Suppose that the vector (𝜔₁,𝜔₂,𝑐)^𝑇 is a solution of the system (73), and for the quasi-periodic wave solution (75), one lets 𝜉_𝑗=𝜉^′_𝑗+𝜏2𝜋𝑖2𝑖^𝑗𝑗,𝜉^′_𝑗=𝑘^′_𝑗𝑥+𝜔^′_𝑗𝑑𝑡+𝜉^(0)′_𝑗, (76) where 𝑘^′_𝑗=2𝜋𝑖𝑘_𝑗,  𝜔^′_𝑗=2𝜋𝑖𝜔_𝑗,  𝜉^(0)′_𝑗=2𝜋𝑖𝜉_𝑗⁽⁰⁾−𝜋𝜏_𝑗𝑗,  and  𝜏₁₂=𝐴₁₂/2𝜋,  𝑗=1,2, and 𝐴₁₂ is given in (21). Then the two-periodic solution (75) tends to the two-soliton solution (21) under a small amplitude limit; that is, 𝑢⟶𝑢₂,𝑎𝑠𝜆₁,𝜆₂⟶0. (77)

Proof. According to formula (67), we expand the function 𝜗(𝜉₁,𝜉₂) in the following form: 𝜗𝜉₁,𝜉₂,𝜏=1+𝑒^{2𝜋𝑖𝜉1}+𝑒^{−2𝜋𝑖𝜉1}𝑒^{−𝜋𝜏11}+𝑒^{2𝜋𝑖𝜉2}+𝑒^{−2𝜋𝑖𝜉2}𝑒^{−𝜋𝜏22},+𝑒^{2𝜋𝑖(𝜉1+𝜉2)}+𝑒^{−2𝜋𝑖(𝜉1+𝜉2)}×𝑒^{−𝜋(𝜏11+2𝜏12+𝜏22)}+⋯. (78) By using (76), we get 𝜗𝜉₁,𝜉₂,𝜏=1+𝑒^𝜉′1+𝑒^𝜉′2+𝑒^{𝜉′1+𝜉′2−2𝜋𝜏12}+𝜆²₁𝑒^−𝜉′1+𝜆²₂𝑒^−𝜉′2,+𝜆²₁𝜆²₂𝑒^{−𝜉′1−𝜉′2−2𝜋𝜏12}+⋯⟶1+𝑒^𝜉′1+𝑒^𝜉′2+𝑒^{𝜉′1+𝜉′2+𝐴12},as𝜆₁,𝜆₂⟶0, (79) where 𝜉^′_𝑗=𝑘^′_𝑗𝑥+∫𝜔^′_𝑗𝑑𝑡+𝜉^(0)′_𝑗,  𝑗=1,2. Thus the two-periodic wave solutions can be reduced to two-soliton solutions (21) under the limit 𝜆₁,𝜆₂→0; we only need to prove that 𝜔^′_𝑗⟶−𝑙(𝑡)𝑘_𝑗^′5−𝑛(𝑡)𝑘^′_𝑗,𝑗=1,2. (80)

And the proof of (80) is similar to the proof of formula (64).

5. Conclusions

In this paper, the generalized variable-coefficient fifth-order Korteweg-de Vries equation is investigated. By virtue of the Bell-polynomial approach, bilinear form of (1) has been derived under condition (2) and bilinear Bäcklund transformation, Lax pairs, and infinite conservation laws of the equation are constructed. Furthermore, the Riemann theta functions have been used to generate one-periodic and two-periodic wave solutions of the equation, and the relations between the periodic wave solutions and soliton solutions are also established. The quasi-periodic solutions play an important role in understanding the diversity and integrability of nonlinear differential equations. We think that there are still many deep relations between the quasi-periodic solutions and other kinds of solutions which still remain open and worth studying.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities (no. 2013XK03) and the National Natural Science Foundation of China (Grant no. 11371361).