Solitary solutions of some nonlinear evolution equations

Abstract

In this paper, the traveling wave reduction combined with the homogeneous balance method (HBM) is used to find new exact solutions by using less-studied nonlinear partial differential equations (nPDG) of higher order. The usual starting point is a special transformation converting the nPDG in its two variables x and t into an nonlinear ordinary differential equation (nODE) in the single variable ξ. Using the hyperbolic tangent method, new exact solutions for the three nPDGs are studied. The new feature of this paper is of course the fact that we are dealing with nPDGs which cannot be found by studying the relevant literature, therefore we believe it is time for publishing some special solutions of this interesting kind of equations. Clearly we do not use a new formalism, but with the aid of the “tanh-method” we are able to calculate at least exact traveling wave solutions (solitary waves) which can physically interpreted as an action of wave propagation. Simultaneous we point out the necessity of such sophisticated methods since a general theory of nPDGs does not exist. The existence of nontrivial, noncomplex valued solutions of the Ramani equation enables us to generate families of real valued solutions of a nonlinear sixth order system. Special differential transformations are leading to connections between traveling wave solutions of associated nonlinear evolution equations.