An Exact Solution to the Combined KS and EdV Equation

The exact solution for the combined KS and KdV equation is obtained via introducing a simple and effective nonlinear transformations.This method is very concise and primary and can be applied to other unintegrable nonlinear evolution equations.It is common knowledge that the Korteweg de Vries(KdV) equation [1] (1)has been proposed as model equation for the weakly nonlinear long waves which occur in many different physical systems; the Kuramoto-Sivashinsky (KS) equationis one of the simplest nonliaear partial differential equations that exhibit Chaotic behavior frequently encounted in the study of continous media [2-4] . Many interesting mathematical and physical properties of eqs. (1) and (2) have been studied widely. But, in several problems where a lonq wavelength oscilatory instability is found, the noulineai evolution of the perturbations near rriticality is governed by the dispersion modified Kuramoto-Sivashi nsky equation(3)ft is clear that this equation is a combination of the KdV and

The exact solution for the combined KS and KdV equation is obtained via introducing a simple and effective nonlinear transformations.This method is very concise and primary and can be applied to other unintegrable nonlinear evolution equations.It is common knowledge that the Korteweg de Vries(KdV) equation [1] (1)has been proposed as model equation for the weakly nonlinear long waves which occur in many different physical systems; the Kuramoto-Sivashinsky (KS) equationis one of the simplest nonliaear partial differential equations that exhibit Chaotic behavior frequently encounted in the study of continous media [2-4] . Many interesting mathematical and physical properties of eqs. (1) and (2) have been studied widely. But, in several problems where a lonq wavelength oscilatory instability is found, the noulineai evolution of the perturbations near rriticality is governed by the dispersion modified Kuramoto-Sivashi nsky equation(3)ft is clear that this equation is a combination of the KdV and KS equaitons. thus eq. (3) is called as the combined KS and KdV equation (CKS-KdVE). Eq. ( 3 ) also appears in problems of fluid flow along an inclined plane [5-7], convection influids with a free surface[8], drift waves in plasmas(9), vertically falling liquid films in the presence of interfacial viscosities[10], etc. Recently, Alfaro and Depassier studied the bifurcation structure of eq. (3)[11]. In this tetter our interest is confined to the determination of exact solutions of eq. (3). Exact solutions of the nonlinear differential equations are specially significant for discussing the related physical problems. Generally, a relevant nonlinear transformation is powerful for solving nonlinear differential equations. In this aspect, tje Cole-Hopf transformation provides an enlightening example.The purpose of this letter is to contribute a new mathematical method to obtain the exact soltion of eq. (3).For the travelling wave solutions u(x, t) = u( ξ ) =u(x-ωt); where is a constant tobe determined, eq. (3) becomesIntegrating the above equatiion with regard to ξ. we obtainwhere λ is an integrating constant. First, leteq. (5) may be wri t ten asVie introduce the following power transformation for eq.(7)where A is a constant to be determined. So that eq. (7) becomesFurther, we make the following crucial ansatzwher a is an undetermined parameter. So we should haveSubstituting eqs. (10)?12) into eq. (9). we obtainBy equating the coefficients of corresponding terms in eq. (13), we obtainwhere b, c and d must satify the following relationIntegrating eg. (10), we havewhere B is an integrating constant. So we obtainIn summary the exact and explicit solitary wave solutions for CKS-KdVE are obtained via introducing the nonlinear transtormatins. We see that the method presented here is indeed simple and effective. The siynificant steps of the about approach are eqs(6),(8)and(10). which are useful trans format ions.We point, out that the same approach can be used to obtain the solitary wave solutions of other related nonlinear equations, such as the nonintegralle KdV -Burgers equation.