Invariant Properties of the Ansatz of the Hirota Method for Quasilinear Parabolic equations

Abstract

We propose a new method based on the invariant properties of the ansatz of the Hirota method which have been discovered recently. This method allows one to construct new solutions for a certain class the dissipative equations classified by degrees of homogeneity. This algorithm is similar to the method of ``dressing'' the solutions of integrable equations. A class of new solutions is constructed. It is proved that all known exact solutions of the FitzHygh-Nagumo-Semenov equation can be expressed in terms of solutions of the linear parabolic equation. This method is compared with the Miura transforms in the theory of Kortveg de Vris equations. This method allows on to create a package by using the methods of computer algebra.