Abstract

Based on the extended homogeneous balance method, the auto-Bcklund transformation transformation is constructed and some new explicit and exact solutions are given for the fourth-order nonlinear generalized Boussinesq water wave equation. Then, the fourth-order nonlinear generalized Boussinesq water wave equation is transformed into the planer dynamical system under traveling wave transformation. We also investigate the dynamical behaviors and chaotic behaviors of the considered equation. Finally, the numerical simulations show that the change of the physical parameters will affect the dynamic behaviors of the system.

1. Introduction

Many nonlinear phenomena can be described by nonlinear wave equations, where Boussinesq equation is one of the important mathematical models, which is widely used in shallow water and the percolation in horizontal porous media [1]. Boussinesq equation has many different forms [2–8]. Clarkson and Kruskal gave a new method for determining similarity reductions of Boussinesq equation in Ref. [2]. Nonclassical symmetry reduction can be used to reduce the Boussinesq equation in Ref. [3]. Based on the asymptotic expansion for small phase speed, an asymptotic semianalytical solution was found in Ref. [4]. Higher-order Boussinesq equations (fourth-order and sixth-order Boussinesq equations ) for two-way propagation of shallow water waves were studied in Ref. [5]. The generalized Boussinesq equation was reduced using Lie group and some exact solutions were obtained in Ref. [6]. The Hirota bilinear method is used to construct two soliton solutions for the ()-dimensional Boussinesq equation, the ()-dimensional Boussinesq equation, and the sixth-order Boussinesq equation. But the three variants of the Boussinesq equation are nonintegrable and do not admit -soliton solutions in Ref. [7]. The generalized ()-dimensional Boussinesq equation is investigated by the bifurcation method of dynamical systems, and some exact solutions were obtained in Ref. [8].

In this work, we consider the fourth-order nonlinear generalized Boussinesq water wave equation: where is a real function and , , and are real nonzero arbitrary constants [9–15].

As far as the authors know, Lie symmetry method is used to analyze Equation (1) and some soliton wave solutions are obtained in Ref. [9]. Based on Hirota’s bilinear method, Qin got some rational solutions and interaction solutions of Equation (1) in Ref. [10]. The rogue wave and semirational solutions of Equation (1) are studied by Hirota’s bilinear method [11]. The general rogue wave solutions of arbitrary orders for Boussinesq equation were found by bilinear Kadomtsev-Petviashvili (KP) reduction method in Ref. [12]. Some travelling wave solutions of Equation (1) were given by extended tanh method and rational method [13, 14]; the solitary wave and shock wave solutions of Equation (1) were obtained by method of undetermined coefficients [15].

Many powerful methods for solving exact solutions of partial differential equations have been developed such as Darboux transform [16, 17] and Lie group method [18–20]. The extended homogeneous balance method (EHBM) is an effective method solving nonlinear partial differential equations [21–23]. So, we apply EHBM to construct new solutions of Equation (1) in this paper.

The analysis of bifurcation and chaos behavior is a very interesting nonlinear phenomenon, which has been applied in many fields, such as engineering, telecommunication, and ecology [24–27]. By analyzing the dynamic behavior of differential equation, we can study whether the periodic external perturbation will lead to the chaotic behavior of differential equation. So, we will study the dynamical behaviors and chaotic behaviors of Equation (1) in this paper.

The rest of this paper is arranged as follows. In Section 2, based on the extended homogeneous balance method, the auto-Bcklund transformation is constructed and some new explicit and exact solutions of Equation (1) are obtained. In Section 3, based on the theory of plane dynamic system, dynamical behaviors and chaotic behaviors of Equation (1) are studied. Some conclusions are provided at the end of the paper.

2. Abundant New Explicit and Exact Solutions to Equation (1)

In terms of the idea of extended homogeneous balance method [21, 22], we assume that the solution of Equation (1) has the following form: where is the arbitrary known seed solution and , , and function are to be determined later.

From (2), we obtain

According to the homogeneous principle method [21, 22], balancing the highest-order derivative term and the highest-order nonlinear term, can be obtained which gives

Thus, (2) can be rewritten as follows

From (6), it is easy to deduce that

Now, substituting (7)–(11) into Equation (1) and simplifying, it yields

Setting the coefficient of the term in Equation (13) to zero, we obtain a nonlinear ordinary differential equation for function :

Integrating Equation (14) once and neglecting the constant of integration gives

Again integrating Equation (15) and letting the constant of integration be zero, we obtain which has particular solution where

According to (17), we get the following results:

By (6) and (17), we obtain the auto-Bcklund transformation of Equation (1) as follows:

Letting the seed solution and using results (19), then Equation (13) can be simplified as

Setting the coefficients of , , , and in Equation (22) to zero yields a set of partial differential equations for as follows:

Here, Equations (24) and (25) can be rewritten as

By analysis of Equations (27) and (28), we find that Equations (23)–(26) are satisfied automatical under the following conditions:

To obtain some exact solutions of Equations (29) and (30), we assume the solutions of the form: where , , , , , , , and are constants which are to be determined.

Substituting (31) into Equations (29) and (30) and simplifying, it leads to a system of nonlinear algebraic equations with respect to , , ,and as follows

By solving the above equations, the following sets of solutions are obtained

Case 1. where

Case 2. where

Case 3. where

Case 4. where

Case 5. where

Case 6. where

Case 7. where

According to (20), (21), and (31), the solution of the fourth-order nonlinear generalized Boussinesq water wave equation is as follows: where

For Case 1, the exact explicit solutions to the fourth-order nonlinear generalized Boussinesq water wave equation are obtained as follows:

If , Equation (34) is reduced to the solitary wave solutions as follows: