The calculation of novel class of solutions of a non-linear fourth order evolution equation by the Weierstraß transform method

摘要

In this paper, the Weierstraß transform method is applied to a non-linear partial differential equation (nPDE) of fourth order to obtain new exact class of solutions in terms of the Weierstraß elliptic function ℘(ξ; g2, g3). The major aspect of this paper however is the fact that we are able to calculate distinctive class of solutions of a less studied non-linear evolution equation, explicitly given by utt + uxxxt + 3uxxuxt = 0. The derived class of solutions cannot be found in the current literature. More precisely, the mathematical structure of these class of solutions are quite different from those derived by classical “ansatz” methods. In other words, using this special algebraic procedure the solution manifold is augmented to further class of solutions including special functions. They are connected in a complicate way and allow a more general representation as found in previous time. Simultaneously we would like to stress the necessity of such sophisticated methods since a general theory of nPDEs does not exist. In addition, this paper is a logical completion of our recent studies where we have applied successfully the Weierstraß transform method to the important KdV–mKdV equation.