Exact and explicit travelling wave solutions for the nonlinear Drinfeld–Sokolov system

Abstract

The Drinfeld–Sokolov (DS) system is investigated by using the tanh method and the sine–cosine method. A variety of exact travelling wave solutions with compact and noncompact structures are formally derived. The study reveals the power of the two schemes in handling identical systems.

Introduction

In this work, we aim to cast light on the Drinfeld–Sokolov system [1], [2], [3]

$$\begin{array}{cc}\hfill & u_t+(v^2{)}_x=0\text{,}\hfill \\ \hfill & v_t-{\mathit{av}}_{\mathit{xxx}}+3{\mathit{bu}}_{x}v+3{\mathit{kuv}}_x=0\text{,}\hfill \end{array}$$

where a, b and k are constants. This system was introduced by Drinfeld and Sokolov as an example of a system of nonlinear equations possessing Lax pairs of a special form [3].

It is well-known that the KdV equation

$$u_t+(u^2{)}_x+u_{\mathit{xxx}}=0$$

describes long nonlinear waves of small amplitude on the surface of inviscid ideal fluid. The KdV Eq. (2) is integrable by the inverse scattering transform and gives rise to solitons. The KdV equation is known to have infinitely many polynomial conservation laws. The KdV Eq. (2) gives rise to solitons, that exist due to the balance between the weak nonlinearity and dispersion of that equation. The term soliton was coined by Zabusky and Kruskal [4], who performed numerical studies of the KdV equation, and found particle like waves which retained their shapes and velocities after collisions.

There has been an enormous number of examples of solitons equations, verifying that the KdV is not just a freak equation [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. Several methods, analytical and numerical, such as Backlund transformation, the inverse scattering method, bilinear transformation, the tanh method [16], [17], [18], the homogeneous balance method, and the sine–cosine ansatz, are used to treat these topics.

The K(n, n) equation

$$u_t+a(u^n{)}_x+(u^n{)}_{\mathit{xxx}}=0\text{,}n>1$$

introduced in [13], gives rise to the so called compactons: solitons with the absence of infinite wings. The delicate interaction between nonlinear convection (uⁿ)_x with genuine nonlinear dispersion (uⁿ)_xxx in the K(n,n) Eq. (3) generates solitary waves with exact compact support that are termed compactons. Unlike the KdV equation, the K(n, n) equations have only a finite number of local conservation laws [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32].

Our main interest of the present work being in implementing two reliable and powerful methods, namely, the sine–cosine method and the tanh method to stress the power of these methods to nonlinear equations. The second aim is the determination of travelling wave solutions with compact and noncompact structures for the Drinfeld–Sokolov (DS) system, a generalized form of the DS system, and a variant of the DS system given by

$$\begin{array}{cc}\hfill & u_t+(v^2{)}_x=0\text{,}\hfill \\ \hfill & v_t-{\mathit{av}}_{\mathit{xxx}}+3{\mathit{bu}}_{x}v+3{\mathit{kuv}}_x=0\text{,}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & u_t+(v^n{)}_x=0\text{,}\hfill \\ \hfill & v_t-{\mathit{av}}_{\mathit{xxx}}+3{\mathit{bu}}_{x}v+3{\mathit{kuv}}_x=0\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & u_t+(v^{-n}{)}_x=0\text{,}\hfill \\ \hfill & v_t-{\mathit{av}}_{\mathit{xxx}}+3{\mathit{bu}}_{x}v+3{\mathit{kuv}}_x=0\hfill \end{array}$$

respectively. Eqs. (4), (5), (6) provide us with the means to meet the primary goals of this work.

As stated before, two strategies will be pursued to achieve our goal, namely, the tanh method [16], [17], [18] and the sine–cosine method [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32]. The systems (4), (5), (6) will be used as testbed for our analysis.