We present a brief report on the different methods for finding exact solutions of nonlinear evolution equations. Explicit exact traveling wave solutions are the most amenable besides implicit and parametric ones. It is shown that most of methods that exist in the literature are equivalent to the "generalized mapping method" that unifies them. By using this method a class of formal exact solutions for reaction diffusion equations with finite memory transport is obtained. Attention is focused to the finite-memory-transport-Fisher and Nagumo equations.
Recently the unified method for finding traveling wave solutions of nonlinear evolution equations was proposed by one of the authors (HIAG). It was shown that, this method unifies all the methods being used to find these solutions. In this paper, we extend this method to find a class of formal exact solutions to Korteweg-de Vries equation with space dependent coefficients.
The eigenvalue problems arise in the analysis of stability of traveling waves or rest state solutions are currently dealt with, using the Evans function method. In the literature, it had been shown that, use of this method is not straightforward even in very simple examples. Here an extended "variational" method to solve the eigenvalue problem for the higher order differential equations is suggested. The extended method is matched to the well known variational iteration method. The criteria for validity of the eigenfunctions and eigenvalues obtained is presented. Attention is focused to find eigenvalue and eigenfunction solutions of the Kuramoto-Slivashinsky and (K[p,q]) equation.
In this paper, we studied the progression of shallow water waves relevant to the variable coefficient Korteweg–de Vries (vcKdV) equation. We investigated two kinds of cases: when the dispersion and nonlinearity coefficients are proportional, and when they are not linearly dependent. In the first case, it was shown that the progressive waves have some geometric structures as in the case of KdV equation with constant coefficients but the waves travel with time dependent speed. In the second case, the wave structure is maintained when the nonlinearity balances the dispersion. Otherwise, water waves collapse. The objectives of the study are to find a wide class of exact solutions by using the extended unified method and to present a new algorithm for treating the coupled nonlinear PDE’s.
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