  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.
