Abstract and Applied Analysis

Volume 2015, Article ID 213847, 7 pages

http://dx.doi.org/10.1155/2015/213847

## New Interaction Solutions of (3+1)-Dimensional KP and (2+1)-Dimensional Boussinesq Equations

Institute of Nonlinear Science, Shaoxing University, Shaoxing 312000, China

Received 22 May 2015; Accepted 6 July 2015

Academic Editor: Leszek Gasinski

Copyright © 2015 Bo Ren et al. 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

The consistent tanh expansion (CTE) method has been succeeded to apply to the nonintegrable (3+1)-dimensional Kadomtsev-Petviashvili (KP) and (2+1)-dimensional Boussinesq equations. The interaction solution between one soliton and one resonant soliton solution for the (3+1)-dimensional KP equation is obtained with CTE method. The interaction solutions among one soliton and cnoidal waves for these two equations are also explicitly given. These interaction solutions are investigated in both analytical and graphical ways. It demonstrates that the interactions between one soliton and cnoidal waves are elastic with phase shifts.

#### 1. Introduction

The investigation of exact solutions of nonlinear partial differential equations (PDEs) plays an important role in the study of nonlinear physical phenomena. Many methods for seeking solutions of PDEs have been developed, such as the inverse scattering method [1], Hirota’s bilinear method [2], symmetry reductions [3], Darboux transformation [4], algebrogeometric method [5], homogeneous balance method [6], and multiple exp-function method [7]. However, except for the soliton-soliton interaction solution, it is very difficult to find interaction solutions among different types of nonlinear excitations with these methods. Recently, a consistent tanh expansion (CTE) method is developed to find interaction solutions between solitons and any other types of solitary waves [8, 9]. The method has been valid for a lot of integrable systems [10–15]. The method for the nonintegrable nonlinear systems is much less studied. In this paper, we use the CTE method to study two typical nonintegrable systems. The interaction solutions between solitons and any other types of solitary waves for these two equations are obtained via the CTE method. These interaction solutions are completely different from those obtained via other methods [6, 16–18].

The structure of this paper is organized as follows. In Section 2, the CTE approach is developed to the (3+1)-dimensional KP equation. The interaction solutions among one soliton and other types of solitary waves such as one resonant soliton solution and cnoidal waves are explicitly given. The interactions between one soliton and cnoidal waves are elastic with phase shifts. According to the above procedure of solving the KP equation, the interaction solutions for the Boussinesq equation are presented in Section 3. The last section is a simple summary and discussion.

#### 2. CTE Method and Interaction Solutions for (3+1)-Dimensional KP System

The (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation reads which has been directly applied in plasma physics. It has also been studied to figure problems of three-dimensional wave structures per se, the wave of collapse of sonic waves, and the self-focusing of the beams of the fast magnetosonic waves propagating in magnetized plasma [19–22].

According to the CTE method, we take the tanh function expansion as the following form by using the leading order analysis [8]: where , , , and are arbitrary functions of . Vanishing the coefficients of powers of , , and after substituting (2) into the (3+1)-dimensional KP system (1), we get Collecting the coefficients of , , , and , we get four overdetermined systems for the field . We omit the expression of these four overdetermined systems since they are very lengthy. According to the CTE approach, we obtain the nonauto-Bäcklund transformation (BT) theorem of (3+1)-dimensional KP equation (1).

*Nonauto-BT Theorem 1*. If one finds the solution to satisfy these four overdetermined systems consistently, then , with will be a solution of (3+1)-dimensional KP system (1).

By means of the nonauto-BT Theorem 1, some special interaction solutions among solitons and other kinds of complicated waves can be obtained. We will give some concrete interesting examples in the following.

A quite trivial straight line solution for the overdetermined systems has the form where , , , and are the free constants. Substituting the trivial solution (5) into (4), one soliton solution of the (3+1)-dimensional KP system yields The nontrivial solution of the (3+1)-dimensional KP equation is found from the quite trivial solution of (5).

To find the interaction solutions between one soliton and other nonlinear excitations, we can use the solutions with one straight line solution (5) plus undetermined waves for the field . For the interaction solution between one soliton and one resonant soliton solution of the (3+1)-dimensional KP equation, we assume where , , , and are arbitrary constants. Substituting expression (7) into overdetermined systems, (7) should be the solution of overdetermined systems with the following relations: Figure 1 displays the interaction behavior between one soliton and one resonant soliton solution with the parameters selected as , , , , , and . This phenomenon can be observed on the sea surface. Solution (7) is useful for applying in maritime security and coastal engineering.