New solitons and periodic solutions for nonlinear physical models in mathematical physics

Abstract

In this paper, we establish exact solutions for three nonlinear equations. The sine–cosine and the exp-function methods are used to construct periodic and soliton solutions of nonlinear physical models. Many new families of exact traveling wave solutions of the nonlinear wave equations are successfully obtained. These solutions may be of significance for the explanation of some practical physical problems. It is shown that the sine–cosine and the exp-function methods provide a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.

Introduction

Recently, the world around us has been inherently nonlinear [1]. Nonlinear evolution equations (NEEs) are widely used as models to describe complex physical phenomena in various fields of sciences, especially in fluid mechanics, solid state physics, plasma physics, plasma wave and chemical physics. Particularly, various methods have been utilized to explore different kinds of solutions of physical models described by nonlinear PDEs. One of the basic physical problems for those models is to obtain their traveling wave solutions. Quite a few methods for obtaining explicit traveling and solitary wave solutions of nonlinear evolution equations have proposed a variety of powerful methods, such as tanh–sech method [2], [3], [4], [5], extended tanh method [6], [7], [8], [9], sine–cosine method [10], [11], [12], homogeneous balance method [13], [14], [15], Jacobi elliptic function method [16], [17], [18], F-expansion method [19], [20], [21] and exp-function method [22], [23], [24]. All these methods have been used to investigate nonlinear dispersive and dissipative problems.

Consider the nonlinear wave equation [25]

$$u_{tt}+k{u}_{xx}+lu+m{u}^3=0\text{,}$$

where $k,l$ and $m$ are constants. Eq. (1.1) contains some particular important equations such as Duffing, Klein–Gordon and Landau–Ginzburg–Higgs equation.

Let us consider the generalized Zakharov equations for the complex envelope $\phi (x,t)$ of the high-frequency wave and the real low-frequency field $v(x,t)$ in the form [26] are:

$$\begin{array}{c}i{\phi }_t+{\phi }_{xx}-2\delta {\left|\phi \right|}^2\phi +2\phi v=0\text{,}\hfill \\ v_{tt}-v_{xx}+{\left({\left|\phi \right|}^2\right)}_{xx}=0\text{,}\hfill \end{array}$$

where the cubic term in (1.2) describes the nonlinear self-interaction in the high-frequency subsystem. This cubic term corresponds to a self-focusing effect in plasma physics. The coefficient $\delta$ is a real constant that can be a positive or negative number.

The (2 +1)-dimensional Davey–Stewartson equations [26] reads

$$\begin{array}{c}i{u}_t+u_{xx}-u_{yy}-2{\left|u\right|}^{2}u-2uv=0\text{,}\hfill \\ v_{xx}+v_{yy}+2{\left({\left|u\right|}^2\right)}_{xx}=0\text{.}\hfill \end{array}$$

This equation which is completely integrable and used to describe the long-time evolution of a two-dimensional wave packet.

Our first interest in the present work is in implementing the sine–cosine method to stress its power in handling nonlinear equations, so that one can apply it to models of various types of nonlinearities. In Section 2, we describe this method for finding exact traveling wave solutions of nonlinear evolution equations. In Sections 3 The exp-function method, 4 The nonlinear wave equation, 5 The generalized Zakharov equations, we illustrate this method in detail with the nonlinear wave, the generalized Zakharov and the (2 +1)-dimensional Davey–Stewartson equations. In Section 6, some conclusions are given.