Advances in Mathematical Physics

Volume 2016, Article ID 4874392, 7 pages

http://dx.doi.org/10.1155/2016/4874392

## CRE Solvability, Exact Soliton-Cnoidal Wave Interaction Solutions, and Nonlocal Symmetry for the Modified Boussinesq Equation

^{1}Department of Mathematics, Yuxi Normal University, Yuxi 653100, China^{2}Ningbo Collaborative Innovation Center of Nonlinear Hazard System of Ocean and Atmosphere, Ningbo University, Ningbo 315211, China

Received 27 February 2016; Accepted 12 May 2016

Academic Editor: Andrei D. Mironov

Copyright © 2016 Wenguang Cheng and Biao Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

It is proved that the modified Boussinesq equation is consistent Riccati expansion (CRE) solvable; two types of special soliton-cnoidal wave interaction solution of the equation are explicitly given, which is difficult to be found by other traditional methods. Moreover, the nonlocal symmetry related to the consistent tanh expansion (CTE) and the residual symmetry from the truncated Painlevé expansion, as well as the relationship between them, are obtained. The residual symmetry is localized after embedding the original system in an enlarged one. The symmetry group transformation of the enlarged system is derived by applying the Lie point symmetry approach.

#### 1. Introduction

The exact solutions for nonlinear evolution equations (NLEEs) arising from many science fields are important because of their wide applications in explaining physical phenomena. Many powerful methods for obtaining the exact solutions of NLEEs have been presented, such as the inverse scattering transformation [1], the Darboux transformations (DT) [2], the Bäcklund transformation (BT) [3], the Lie group method [4, 5], Hirota’s bilinear method [6], the Painlevé analysis [7], separated variable method [8], homogeneous balance method [9], and tanh function expansion method [10]. For these methods, the interaction solutions among different nonlinear excitations such as solitons on a cnoidal wave background are difficult to obtain. The soliton solutions on the background of the periodic solutions which are given by the elliptic theta function can be found with the help of Cauchy-Baker kernel [11, 12].

Recently, Lou found that the symmetry from the truncated Painlevé expansion is just the residue with respect to the singular manifold and proposed residual symmetry [13, 14]. Furthermore, by developing the truncated Painlevé expansion, Lou [15, 16] established the consistent Riccati expansion (CRE) method, which is a simple but effective method to construct interaction solutions among different nonlinear excitations. The method is valid for some nonlinear models like the bosonized supersymmetric Korteweg-de Vries (KdV) model [14], the nonlinear Schrödinger system [16], the Broer-Kaup System [17], and so on [18–22].

In this paper, we focus on investigating the CRE solvability and nonlocal symmetry of the modified Boussinesq equation:which was proposed by Hirota and Satsuma [23] from a BT of the Boussinesq equation:Equation (1) is called the modified Boussinesq equation since it is linked with the known Boussinesq equation (2) by the Miura transformation:The generalized Hamiltonian form and the corresponding finite-dimensional integrable systems of (1) are obtained with the help of the nonlinearization approach of Lax pairs [24]. The Painlevé property [25] and similarity solutions [26, 27] have been studied. To our knowledge, the analytic interaction solutions among different nonlinear excitations for (1) have not been obtained up to now. Here we will study the interaction solution between the soliton and the cnoidal periodic wave of (1).

The paper is arranged as follows. Section 2 is devoted to the CRE method for the modified Boussinesq equation. As a result, two types of special explicit interaction solution between the soliton and the cnoidal periodic wave of this equation are given. In Section 3, for the modified Boussinesq equation, we derive the nonlocal symmetry related to the CTE and the nonlocal residual symmetry from the truncated Painlevé expansion. The relationship between them is presented. Then the corresponding symmetry group transformation is found in the process of the localization of residual symmetry. The last section is a short summary and discussion.

#### 2. CRE Solvability and Soliton-Cnoidal Wave Interaction Solutions

##### 2.1. CRE Solvability

Based on the CRE method in [15], the possible truncated expansion expression for (1) has the formwhere , , and are undetermined functions of space-time and and satisfies the Riccati equationwhich includes as a special case. Differentiating with respect to , (1) becomes

Substituting (4) with (5) into (6) and vanishing all the coefficients of the powers of , we obtain ten overdetermined equations for only three undetermined functions. Fortunately, these overdetermined equations are consistent and possess the following solution:and the function satisfies a generalization of the Schwartzian form of (1):where

Due to consistency of the overdetermined system, we call the modified Boussinesq equation CRE solvable.

##### 2.2. CTE Solvability and Soliton-Cnoidal Wave Interaction Solutions

We consider the function in Riccati equation (5) as the following special solution:which means that a CRE solvable system must be consistent tanh expansion (CTE) solvable, and vice versa. Meanwhile, truncated expansion expression (4) is changed aswhere , , and are undetermined by (7) and (8) with , , , and . Thus, we haveand the function satisfies

In brief, we arrive at the following nonauto-BT theorem for (1).

Theorem 1. *If is a solution of (13), then , withis a solution of modified Boussinesq equation (1).*

Theorem 1 shows that the single soliton solution of (1) is only the straight line solution of (13); the interaction solutions between solitons and other nonlinear excitations of (1) can be constructed by solving (13). To find the interaction solutions between one soliton and other nonlinear waves of (1), we consider in the formwhere is a function with respect to and . In this study, we only discuss the solutions with the form

Substituting (16) into (13), we can find that satisfieswithand , are arbitrary constants. Then (1) has the explicit solution expressed as

It is clear that (17) has abundant explicit solutions in terms of Jacobi elliptic functions. Hence, solution (19) exhibits the interactions between one soliton and cnoidal periodic waves. In what follows, only two nontrivial cases are considered in detail to obtain this kind of solution.

*Case 1. *A simple solution of (17) is given aswhich leads to the soliton-cnoidal wave interaction solution of (1) as follows:where are arbitrary constant, , , , andIn solution (21), is the third type of incomplete elliptic integral.

The dynamic behavior of soliton-cnoidal wave interaction solution (21) with the parameters , , , and is illustrated in Figure 1. It can be seen from Figure 1 that a soliton moves on a cnoidal wave background instead of moving on the plane continuous wave background. This kind of solution can be applicable to describe some interesting physical phenomena, such as the Fermionic quantum plasma [28].