Explicit Traveling Waves and Invariant Algebraic Curves

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.