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

and Applied Analysis 3 in which n 1 + n 2 + ⋅ ⋅ ⋅ + n l ≥ 1 and operators D x 1 ⋅ ⋅ ⋅ D x l are classical Hirotas bilinear operators defined by D n 1 x 1 ⋅ ⋅ ⋅ D n l x l F ⋅ G = (∂ x 1 − ∂ x 󸀠 1 ) n 1 ⋅ ⋅ ⋅ (∂ x l − ∂ x 󸀠 l ) n l × F (x 1 , . . . , x l ) × G(x 󸀠 1 , . . . , x 󸀠 l ) 󵄨 󵄨 󵄨 󵄨 󵄨x 󸀠 1 =x 1 ,...,x 󸀠 l =x l . (10) In the particular case, when F = G, formula (9) becomes G −2 D n 1 x 1 ⋅ ⋅ ⋅ D n l x l G ⋅ G = Y n 1 x 1 ,...,n l x l (0, q = 2 lnG) ={ 0, n 1 +⋅ ⋅ ⋅+n l is odd , P n 1 x 1 ,...,n l x l (q) , n 1 +⋅ ⋅ ⋅+n l is even , (11) in which the P-polynomials can be characterized by an equally recognizable even part partitional structure P 2x (q) = q 2x , P x,t (q) = q xt , P 4x (q) = q 4x + 3q 2 2x , P 6x (q) = q 6x + 15q 2x q 4x + 15q 3 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 theY-polynomials, then it can be easily transformed into the corresponding bilinear Bäcklund transformation form. Theorem 2 (see [11]). The binary Bell polynomials Y n 1 x 1 ,...,n l x l (V, w) can be separated into P-polynomials and Y-polynomials: (FG) −1 D n 1 x 1 ⋅ ⋅ ⋅ D n l x l F ⋅ G = Y n 1 x 1 ,...,n l x l (V, w)|V=lnF/G,w=lnFG = Y n 1 x 1 ,...,n l x l (V, V + q)󵄨󵄨󵄨 󵄨V=lnF/G, q=2 lnG


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][3][4] and the Bäcklund transformation [5,6], the Hirota method [7,8], and algebrogeometrical approach [9][10][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][13][14].Fan developed this method to find infinite conservation laws of soliton equations [15][16][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 variablecoefficient KdV equation, for example, the nonlinear excitations of a Bose gas of impenetrable bosons with longitudinal 2 Abstract and Applied Analysis confinement, the nonlinear waves in types of rods [28][29][30].Obviously, equations with variable-coefficient are much more complicated than constant-coefficient forms, and much attention has been paid to this subject [31][32][33][34][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: where  is a function of  and  and (), (), (), (), (), (), (), and () are analytic functions of .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 variablecoefficient models mapped to the completely integrable constant-coefficient counterparts [38], will be where  ̸ = 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 twoperiodic 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.

Binary Bell Polynomials
To begin with, we will give some basic concepts and notations about the Bell polynomials.For details, please refer to [11][12][13].
Let  = ( 1 ,  2 , . . .,   ) be a  ∞ function with multivariables; the following polynomials are called the multidimensional Bell polynomials, where For convenience, we denote the multidimensional Bell polynomials by -polynomials.
For example, for the simplest case  = (), the onedimensional Bell polynomials are For  = (, ), the two-dimensional Bell polynomials are Based on the above Bell polynomials, the multidimensional binary Bell polynomials (Y-polynomials) can be defined as follows: The Y-polynomials inherit the easily recognizable partial structure of the Bell polynomials.The first few lowest order binary Bell polynomials are Theorem 1 (see [11]).The link between binary Bell polynomials Y  1  1 ,...,    (V, ) and the standard Hirota bilinear equation ⋅  can be given by an identity Abstract and Applied Analysis 3 in which  1 +  2 + ⋅ ⋅ ⋅ +   ≥ 1 and operators   1 ⋅ ⋅ ⋅    are classical Hirotas bilinear operators defined by In the particular case, when  = , formula (9) becomes in which the -polynomials can be characterized by an equally recognizable even part partitional structure 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.

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.

Bilinear Representation.
In order to detect the existence of the bilinear representation, we introduce a potential field  by setting with  = () being a free function with respect to , which will be chosen appropriately so that ( 1) is related to the polynomials.Then substituting ( 15) into (1) and integrating with respect to  and noting condition (2) yield Comparing the fourth and the sixth terms of the above equation with formula (12) implies that we should require () =  − ∫ () .The resulting equation is then cast into a combination form of the -polynomials: Making a change of the dependent variable and noting property (11), we can obtain the bilinear representation of (1) as Following the Hirota bilinear theory, one-soliton solution for (1) in explicit forms can be given as and two-soliton solutions are denoted by with where   and  0  ,  = 1, 2, are arbitrary real constants.

Bäcklund Transformation and Lax Pair.
In the following, we search for the bilinear Bäcklund transformation and the Lax pair of (1).Let   = 2 ln  and  = 2 ln  be two different solutions of (17), respectively; we have the two-field condition If set then ( 24) can be rewritten as with Taking where  is an arbitrary parameter.Then from (28), (V, ) can be rewritten in the form Then from ( 26)-( 29), we deduce a coupled system of Ypolynomials: By application of the identity (13), the system (30) immediately leads to the following bilinear Bäcklund transformation: where  is an arbitrary parameter.By using the Hopf-Cole transformation V = ln , it follows from formulas ( 13) and ( 14) that It is easy to check that the integrability condition is satisfied if  2 = ( ∫ () /) and  is a solution of the generalized variable-coefficient fifth-order Korteweg-de Vries (1).

Infinite Conservation Laws.
Next, through the Bellpolynomial-type Bäcklund transformation, we will perform the procedure of deriving the infinite sequence of conservation laws of (1) in the following form: Let it follows from relation (25) that Rewrite (30) in the conserved form Substituting ( 37) into (38), we can obtain where we have used (39) to get (40).
To proceed, inserting the expansion into (39) and equating the coefficients for power of , we then obtain the recursion relations for the conserved densities   : and the recursion relation is given as In addition, substituting (41) into (40) yields  With the recursion formulae of   and   presented previously, the infinite conservation laws for (1) can be constructed.

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 : Here the integer value vector  = ( 1 , . . .,   )  ∈   , and complex phase variables  = ( 1 , . . .,   )  ∈   .Moreover, for two vectors  = ( 1 , . . .,   )  and  = ( 1 , . . .,   )  , their inner product is defined by The  = (  ) is a positive definite and real-valued symmetric  ×  matrix, which we call the period matrix of the theta function.The entries   of the period matrix  can be considered as free parameters of the theta function (45).Now, we consider the solution for (1) in the following bilinear form: where  is the constant of integration.

Construction of One-Periodic Waves.
In this section, we consider the one-periodic wave solutions for (1).When  = 1, the theta function reduces the following Fourier series in : where the phase variable  =  + ∫ +  (0) and the parameter  > 0.
and Applied Analysis 7 Substituting (48) into (47), we obtain where By shifting sum index as  =   + 1, we conclude that which imply that if (0) = (1) = 0, then it follows that and thus the theta function (48) is the exact solution of (47).
In this way, we may let Denote Then (53) can be written as Notice that there are a lot of choices for the angular wave number .The determinant of the coefficient matrix () = (  ()) 2×2 is a polynomial in ; if det(()) ̸ ≡ 0, then is either an empty set or a finite set, and so, there are real solutions (, ) to the system (55) for  ∉  0 .Solving this system, we have Therefore we get a one-periodic wave solution of (1): where the parameter  is given by (57).

Property of One-Periodic Waves.
In the following, we further consider asymptotic properties of the oneperiodic 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 (, )  is a solution of the system (55), and for the quasi-periodic wave solution (58), we let where   = 2,   = 2,  (0)  and = 2 (0) − .Then the one-periodic solution (58) tends to the one-soliton solution (20) under a small amplitude limit; that is, Proof.By using (54), we write functions   ,   , ,  = 1, 2, as the series about : Suppose that the solution of system (55) has the following form: substituting expansions (61) and (62) into system (55), and let  → 0; we can obtain the following relation immediately: (64) It remains to show that the one-periodic wave (58) degenerates to the one-soliton solution (20) under the limit  → 0.

Asymptotic Property of Two-Periodic Waves.
In this subsection, we consider the asymptotic properties of the twoperiodic 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.