Explicit Traveling Waves and Invariant Algebraic Curves

TL;DR

This paper defines algebraic traveling wave solutions for n-th order PDEs and proves their existence is equivalent to the presence of invariant algebraic curves in associated ODE systems, with applications to the Fisher-Kolmogorov equation.

Contribution

It introduces a precise definition of algebraic traveling waves and establishes a novel equivalence with invariant algebraic curves in related ODEs.

Findings

All known explicit traveling waves are algebraic solutions.

Existence of algebraic traveling waves is characterized by invariant algebraic curves.

For the Fisher-Kolmogorov equation, only known solutions are algebraic.

Abstract

In this paper we introduce a precise definition of algebraic traveling wave solution for general n-th order partial differential equations. All examples of explicit traveling waves known by the authors fall in this category. Our main result proves that algebraic traveling waves exist if and only if an associated n- dimensional first order ordinary differential system has some invariant algebraic curve. As a paradigmatic application we prove that, for the celebrated Fisher- Kolmogorov equation, the only algebraic traveling waves solutions are the ones found in 1979 by Ablowitz and Zeppetella. To the best of our knowledge, this is the first time that this type of results have been obtained.