Compactons and topological solitons of the Drinfel ’ d – Sokolov system

Abstract. This paper presents the Drinfel’d–Sokolov system (shortly D(m,n)) in a detailed fashion. The Jacobi’s elliptic function method is employed to extract the cnoidal and snoidal wave solutions. The compacton and solitary pattern solutions are also retrieved. The ansatz method is applied to extract the topological 1-soliton solutions of the D(m,n) with generalized evolution. There are a couple of constraint conditions that will fall out in order to exist the topological soliton solutions.


Introduction
In 1834, Scott Russell discovered the solitary wave phenomena.Researching for solitary wave solution in nonlinear mathematical physics has been an significant topic.A class of solitary waves with compact support, called compactons by Rosenau and Hyman [1], was inspected.Compactons were shown to knock elastically and also disappeared in finite nuclei of exterior region.In recent years, many powerful methods had been proved such as the homogeneous balance method [2], the hyperbolic function expansion method [3], the Jacobi elliptic function method [4], F -expansion method [5], homotopy analysis method [6], the bifurcation theory method of dynamical system [7] and Weierstrass elliptic function method [8].In this paper, we consider the nonlinear dispersion D(m, n) [9], c Vilnius University, 2014 where a, b, c, m, n are parameters.The nonlinear D(m, n) was exposed as a model of water waves.Xie and Yan [9] studied system (1) and they obtained many types of compacton and solitary pattern solutions.Deng et al. [10] obtained more new exact travelling wave solutions of system (1) by using the Weierstrass elliptic function method.Zhang et al. [11] investigated some smooth and non-smooth travelling wave solutions of system (1) by using the bifurcation theory of planar dynamical system.Sweet et al. [12] obtained trigonometric and hyperbolic type solutions to (1) by using the homotopy analysis method.Sweet et al. [13] worked gDS equation by using a Miura-type transformation.Wen et al. [14] investigated some explicit expressions of solutions for the classical Drienfel'd-Sokolov-Wilson equation (DSWE) (1) by using the bifurcation method and qualitative theory of dynamical system Equation has been studied by [15,16].Biswas and Triki [17] worked the 1-soliton solution of Eq. ( 1) with power law nonlinearity by using the solitary wave ansatz method.The paper is organized as follows.In Section 2, a reduction of Eq. ( 1) is made and the concrete scheme of the approach for solving the equation is presented.New type Jacobi elliptic functions solutions, compactons, solitary pattern and travelling wave solutions of the equation D(m, n) are obtained in Section 3. In Section 4, we get the topological soliton solution of this system by using the ansatz method.Some conclusions are given in Section 5.

The solutions to the D(m, n) system
We first consider a nonlinear partial differantial equation of the form where F is a polynomial function with respect to variable changeable or some function which can be reduced to a polynomial function by using some transformations.
Here the developed Jacobi elliptic function method introduces many new traveling wave solutions.We set the solutions in the form where A, B, β are parameters that will be determined, sn, cn, sc, nc are the Jacobi elliptic functions and is the modulus of the Jacobi elliptic functions (0 < < 1).Let where k, λ are constants to be determined.
Integrating the first equation of system (6) with respect to ξ, yields where C is a integration constant [9].Substituting (7) into the second equation of system (6) and integrating once, we get where C 0 is an integration constant and we accept C 0 = 0.

Exact soliton and solitary pattern solutions of nonlinear D(m, n) equation
To work soliton and solitary pattern solutions of (1), we deal with another transformation If we substitution (35) into ( 8), then we have the following solutions.
(iii) Many solutions that obtained by Xie and Yan [9] by using sine-cosine method are gotten by Jacobi elliptic function method which we have present in this study.
including the cnoidal and snoidal wave solutions were obtained.The compacton, solitary pattern, periodic wave solutions were also obtained.Finally, the topological 1-soliton solution was also retrieved for the D(m, n) equation with genreralized evolution.In this case a couple of constraint conditions fell out during the course of derivation of the soliton solution that must hold in order to exist the solution.