A novel class of solutions for a non-linear third order wave equation generated by the Weierstraß transformation

Abstract

Abstract In this paper, a traveling wave reduction combined with the transformation method in terms of Weierstras elliptic functions is used to find a class of new exact solutions for a non-linear partial differential equation (nPDE) of third order, the so called combined KdV–mKdV equation. The usual starting point is a special transformation (the traveling wave “ansatz”) converting the nPDG in its two variables x and t to the belonging non-linear ordinary differential equation (nODE) in the single variable ξ. Using the Weierstras elliptic-function method, new exact class of solutions in terms of the function ℘(ξ; g2, g3) are obtained. Moreover, class of solutions showing typical solitary behavior results as a special case. The important aspect of this paper however is the fact, that we are able to calculate distinct class of solutions which cannot be found in current literature. In other words, using this method, the solution manifold is augmented to new class of solution functions. In the same time we would like to stress the necessity of such sophisticated methods since a general theory of nPDEs does not exist at present.