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

TL;DR

This paper introduces a novel method leveraging invariant properties of the Hirota ansatz to construct new solutions for dissipative quasilinear parabolic equations, enhancing solution generation and computational approaches.

Contribution

It presents a new solution construction method based on invariants of the Hirota ansatz, applicable to a class of dissipative equations, and compares it with existing techniques like Miura transforms.

Findings

Constructed new solutions for dissipative equations.

Showed all known solutions of FitzHugh-Nagumo-Semenov can be expressed via linear parabolic solutions.

Demonstrated the method's compatibility with computer algebra systems.

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.