The extended tanh method for the Zakharov–Kuznetsov (ZK) equation, the modified ZK equation, and its generalized forms

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Abstract

The reliable extended tanh method, that combines tanh with coth, is used for analytic treatment of the Zakharov–Kuznetsov (ZK) equation, the modified ZK equation, and the generalized forms of these equations. New travelling wave solutions with solitons and periodic structures are determined. The power of the employed method is confirmed.

Introduction

The Kadomtsov–Petviashivilli (KP) equation, and the Zakharov–Kuznetsov (ZK) equation given by{ut+auux+uxxx}x+uyy=0,andut+auux+buxx+uyyx=0,respectively are the best known two-dimensional generalizations of the KdV equation investigated in [1], [2], [3], [4], [5], [6], [7], [8], [9] by many distinct approaches. The KP equation, introduced in [7], is integrable and describes the evolution of quasi-one-dimensional shallow-water waves when effects of the surface tension and the viscosity are negligible. The ZK equation, presented in [8], governs the behavior of weakly nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field [1], [2].

The ZK equation, which is a more isotropic two-dimensional, was first derived for describing weakly nonlinear ion-acoustic waves in a strongly magnetized lossless plasma in two dimensions [8]. Unlike the KP equation, the ZK equation is not integrable by the inverse scattering transform method. It was found that the solitary wave solutions of the ZK equation are inelastic.

Also in the context of plasma physics, Schamel [3] derived the (2+1)-dimensional equationsut+u1/2ux+b(uxx+uyy)x=0,andut+(1+bu1/2)ux+12uxxx=0,that describe ion-acoustic waves in a cold-ion plasma but where the electrons do not behave isothermally during their passage of the wave. Munro and Parkes [1], [2] showed that if the electrons are non-isothermal, the governing equation of the ZK equation is a modified form, refereed to as the mZK equation. They also showed that with an appropriate modified form of the electron number density proposed by Schamel [3], a reductive procedure leads to a modified form of the ZK equation, namely16ut+20(u3/2)x+(uxx+uyy)x=0,

Motivated by the rich treasure of the ZK equation and its modified forms in the literature of the nonlinear development of ion-acoustic waves in a magnetized plasma, an analytic study will be conducted on the ZK equation:ut+auux+b(uxx+uyy)x=0,and the generalized ZK equation (gZK)ut+aunux+b(uxx+uyy)x=0,n1.Moreover, we will study and the modified ZK equation (mZK)16ut+30u1/2ux+(uxx+uyy)x=0,and its generalized formut+aun/2ux+b(uxx+uyy)x=0,n1.Li et al. [6] examined the Zakharov–Kuznetsov equation and obtained exact travelling wave equations by using the extended tanh method and direct assumption method. As a result of study in [6], kink-shaped and bell-shaped solitons were formally derived. For more details about the solitary wave solutions and the ZK equation, the reader is advised to read [10], [11], [12], [13], [14], [15], [16]. More works can be found about solitons, kinks and compactons can be found in [17], [18], [19], [20]. In this work we will employ the reliable extended tanh method for analytic treatment of these equations.

The objectives of this work are twofold. First, we seek to establish exact solutions of distinct physical structures, solitons and periodic wave solutions, for the nonlinear dispersive Eqs. (6), (7), (8), (9). Second, we aim to implement the extended tanh method [10], [11], [12], [13] to obtain new exact travelling wave solutions. The power of this method is investigated and confirmed. In what follows, the method will be reviewed briefly.

Section snippets

The extended tanh method

A PDEP(u,ut,ux,uxx,uxxx,)=0,can be converted to an ODEQ(u,u,u,u,)=0,upon using a wave variable ξ = (x  ct). Eq. (11) is then integrated as long as all terms contain derivatives where integration constants are considered zeros.

The standard tanh method is developed by Malfliet [10], [11] where the tanh is used as a new variable, since all derivatives of a tanh are represented by a tanh itself. Introducing a new independent variableY=tanh(μξ),ξ=x-ct,leads to the change of derivatives:ddξ=μ(1-Y2)d

The Zakharov–Kuznetsov (ZK) equation

In this section we employ the extended tanh method to the Zakharov–Kuznetsov equation in the (2 + 1), two spatial and one time, dimensionsut+auux+b(uxx+uyy)x=0,where a and b are constants. Using the wave variable ξ = x + y  ct carries the PDE (15) into the ODE-cu+a2(u2)+2bu=0,where by integrating (16) and neglecting the constant of integration we obtain-cu+a2u2+2bu=0.Balancing u″ with u2 in (17) givesM+2=2M,so thatM=2.The extended tanh method (14) admits the use of the finite expansionu(μξ)=S(Y)=k

The generalized ZK equation

In this section we study the generalized ZK equationut+aunux++b(uxx+uyy)x=0,n>1.Using the wave variable ξ = x + y  ct in (39) and integrating the resulting equation we find-cu+an+1un+1+2bu=0,n>1.

Balancing un+1 with u″, we find(n+1)M=M+2,so thatM=2n.To get a closed form solution, M should be an integer. To achieve our goal, we use the transformationu(ξ)=v1n(ξ),that will carry (40) into the ODE-cn2(n+1)v2+an2v3+2bn(n+1)vv-2b(n2-1)(v)2=0.

Balancing v″ with v2 in (44) gives M = 2. The extended tanh

The modified ZK equation (mZK)

As stated before, with appropriate scalings [1], [2], the Zakharov–Kuznetsov equation may be transformed to the modified form16ut+20(u3/2)x+uxx+uyyx=0.Using the wave variable ξ = x  ct carries the PDE (55) into the ODE-16cu+20(u3/2)+2u=0,where by integrating (56) and neglecting the constant of integration we obtain-16cu+20(u3/2)+2u=0.Balancing u3/2 with u″ gives M = 4. The extended tanh method assumes that the solution can be expressed by the expansionu(μξ)=S(Y)=k=02akYk+k=12bkY-k2,

Generalized form of modified ZK equation (gmZK)

We close this work by studying a generalized form of modified ZK equation (gmZK) given byut+aun2ux+b(uxx+uyy)x=0,n1,that can be reduced to the ODE-cu+2an+2un+22+2bu=0,obtained upon using the wave variable ξ = x  ct and by integrating the resulting equation. Balancing un+22 with u″ gives M=4n. To get a closed form solution, M should be an integer. Hence, we use the transformationu(ξ)=v2n(ξ),so that Eq. (70) will be carried into the ODE-cn2(n+2)v2+2an2v3+4bn(n+2)vv-4b(n2-4)(v)2=0.

Balancing v3

Discussion

In this paper, we exhibited the extended tanh method to study the Zakharov–Kuznetsov equation, the generalized ZK equation, the modified ZK equation and a generalized form of the modified ZK equation. New solitons and periodic solutions were formally derived. These solutions may be helpful to describe waves features in plasma physics. Moreover, the obtained results in this work clearly demonstrate the reliability of the extended tanh method. We confirm the power of the (extended tanh method,

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