New sets of solitary wave solutions to the KdV, mKdV, and the generalized KdV equations

Abstract

In this work we introduce new schemes, each combines two hyperbolic functions, to study the KdV, mKdV, and the generalized KdV equations. It is shown that this class of equations gives conventional solitons and periodic solutions. We also show that the proposed schemes develop sets of entirely new solitary wave solutions in addition to the traditional solutions. The analysis can be used to a wide class of nonlinear evolutions equations.

Introduction

This paper is concerned with the KdV, mKdV, and the generalized KdV equations given by

$$u_t+{\mathit{auu}}_x+{\mathit{bu}}_{3x}=0\text{,}$$

$$u_t+{\mathit{au}}^2{u}_x+{\mathit{bu}}_{3x}=0\text{,}$$

and

$$u_t+{\mathit{au}}^n{u}_x+{\mathit{bu}}_{3x}=0\text{,}n⩾1\text{,}$$

respectively, with constant parameters a and b, and $u_{\mathit{rx}}=\frac{{\mathrm{\partial }}^r}{\mathrm{\partial }{x}^r}$. The class of the KdV equations is a typical system where the stable particle-like waves, i.e solitons, arise due to the balance between dispersion u_xxx and nonlinearity uⁿu_x, n ⩾ 1.

A great deal of research work has been invested during the past decades for the study of the nonlinear dispersive KdV equation [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13] and many forms of KdV-like equations. The physical structures of the nonlinear dispersive equations have been investigated by using various analytical and numerical methods. A variety of powerful methods, such as the inverse scattering method [6], Wadati trace method [1], [2], Backlund transformation method, Hirota bilinear forms and many other methods, were used to study nonlinear dispersive and dissipative models. The tanh method, developed by Malfliet [9], and used in [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20] among many others, is heavily used in the literature to handle nonlinear evolutions equations. Fan et al. [7] combined the standard tanh method with the Riccati equation to introduce a useful extension to the tanh method. The extension worked effectively in nonlinear models and is used by many researchers. Recently, a set of ansatze that involve trigonometric and hyperbolic functions is introduced in [19], [20], [21], [22] to develop new solutions to nonlinear evolution equations in general.

Our first interest in this work is implementing new strategies that involve hyperbolic functions. The strategies that will be pursued in this work rests mainly on combining two of the hyperbolic functions in specific ways. The next goal is the determination of new solitary wave solutions. The proposed schemes, as we believe, are entirely new and introduce new solutions in addition to the well-known traditional solutions. The ease of using these methods, to determine shock or solitary type of solutions, shows its power.