On numerical doubly periodic wave solutions of the coupled Drinfel’d–Sokolov–Wilson equation by the decomposition method

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Abstract

In this paper, we consider obtaining approximate doubly periodic wave solutions of the coupled Drinfel’d–Sokolov–Wilson equation by using the Adomian decomposition method (ADM). Then we obtain the numerical Jacobi elliptic function solution for the initial conditions. The numerical results are compared with the known exact doubly periodic solutions. Most of the symbolic and numerical computations have been performed using the Computer Algebra System—Mathematica.

Introduction

Over the last 20 years, the Adomian decomposition approach has been applied to obtain formal solutions to a wide class of both deterministic and stochastic PDEs. This method has some significant advantages over numerical methods in that it provides analytic, verifiable and rapidly convergent approximation which yields an insight into the character and behaviour of the solution just as in a closed form solution [1], [2], [3], [4], [5].

Nonlinear partial differential equations (NPDEs) are widely used to describe complex phenomena in various sciences, especially in physical sciences. Finding explicit and exact solutions, in particular, solitary wave solutions of nonlinear evolution equations in mathematical physics plays an important role in nonlinear science. It is well known that there are infinite solutions for every NPDE, and it is a difficult task to find an exact solution. In various fields of science and engineering, nonlinear evolution equations, as well as their analytic solutions, are of fundamental importance. Analytic and soliton solutions of such equations are either not available or obtained using transformations based on, for example, one parameter Lie-Bäcklund groups analysis [6], invariance group analysis [7], Lie infinitesimal criterion [8], the inverse scattering transform [9], the symbolic computations [10] and the Bäcklund transformation [11]. A feature common to all these methods is that they use transformations to reduce the equation into more simple equations and then solve it. Unlike classical techniques, nonlinear equations are solved easily and elegantly without transforming the equation by using ADM. The technique has many advantages over the classical techniques; it mainly avoids linearization and perturbation in order to find solutions of given nonlinear equations. It provides an efficient analytic and soliton solutions with a high accuracy, minimal calculation, avoidance of physically unrealistic assumptions.

The convergence of this method was proved by Cherruault and co-operations. In [12], Cherruault proposed a new definition of the method and he then insisted that it will become possible to prove the convergence of the decomposition method. In [13], Cherruault and Adomian proposed a new convergence proof of Adomian’s method based on properties of convergent series. In [14], a new approach of the decomposition method was obtained in a more natural way than was given in the classical presentation. In addition, this paper also includes a new condition for obtaining convergence of the decomposition series.

In this paper, we consider the coupled Drinfel’d–Sokolov–Wilson equation in [15] with initial conditions in the following form:ut+3vvx=0,,vt+2vxxx+2uvx+uxv=0,with the initial conditions u(x, 0) = f(x) and v(x, 0) = g(x). This equation was introduced as a model of water waves [16], [17]. The solitary wave solution and solitary wave structure of this system were investigated [18].

In this paper, we investigate the ADM for obtaining Jacobi elliptic approximate solutions of the coupled Drinfel’d–Sokolov–Wilson equation. We also give a framework for the theoretical analysis of the ADM. The decomposition method will be illustrated by studying the coupled Drinfel’d–Sokolov–Wilson equation to compute explicit and numerical solutions.

This paper is organized as follows: In the next section, we introduce a framework for theoretical analysis of the ADM for the coupled Drinfel’d–Sokolov–Wilson equation. In Section 3, we study one example for some values of the constants. In Section 4, we discuss numerical implementations and present some numerical results which agree well with the theoretical analysis and which demonstrate the effectiveness of this approach. The last section contains some remarks and conclusions.

Section snippets

Analysis of the ADM

In this section, we describe the algorithm of the ADM as it applies to the coupled Drinfel’d–Sokolov–Wilson equation. For this we write Eq. (1) in the following operator form:Ltu=-3vvx,Ltv=-2vxxx-2uvx-uxv.Following [1], [2], [3], [4], [5] we define for Eq. (1) the linear operator Lt=t and Lx=3x3. By defining the onefold right-inverse operator Lt-1=0t(.)dt, we can formally getu(x,t)=u(x,0)-3Lt-1(vvx),v(x,t)=v(x,0)-Lt-1[2vxxx+2uvx+uxv].Therefore,u(x,t)=f(x)-3Lt-1[N(u,v)],v(x,t)=g(x)-Lt-1[2Lxv

Exemplification of the ADM

We consider the application of the ADM to the coupled Drinfel’d–Sokolov–Wilson equation (1) with the initial conditionsu(x,0)=-3k2qds2[kx,m],v(x,0)=2ckqdskx,m,where k, q and c arbitrary constants, and m is the modulus of the Jacobi elliptic functions (0 < m < 1). In addition the ds[kx,m] is the Jacobi elliptic function.

Using (11) with (7), (8), (9) for the coupled equation (1) and initial conditions (13) givesu0=-3k2qds2[kx,m],u1=6k2cqcs[kx,m]ds[kx,m]ns[kx,m]t,u2=-3(k2q(c2cs2[kx,m]ns2[kx,m]+c2ds[kx,

Numerical results

In this section, we consider the coupled Drinfel’d–Sokolov–Wilson equation for numerical comparison. In order to verify numerically whether the proposed methodology leads to a higher accuracy, we can evaluate the approximate solution using the n-term approximation (12). It is to be noted that ϕn and ϕn clearly shows the convergence to the correct limit is presented. Table 1, Table 2, Table 3, Table 4 show the difference of the exact and numerical solutions of the absolute errors. The graphs of

Conclusions

In this paper, the Adomian decomposition method was used for obtaining an approximate Jacobi elliptic solution of the coupled Drinfel’d–Sokolov–Wilson equation with initial conditions using the PC-based Mathematica package for an illustrated example. The numerical results showed that this method has great accuracy and reductions of the size of calculations compared with the other methods. It may be concluded that the Adomian methodology is a very powerful and efficient technique in finding

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