Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem
Abstract
We consider reaction-diffusion equations and Korteweg-de Vries-Burgers (KdVB) equations, i.e. scalar conservation laws with diffusive-dispersive regularization. We review the existence of traveling wave solutions for these two classes of evolution equations. For classical equations the traveling wave problem (TWP) for a local KdVB equation can be identified with the TWP for a reaction-diffusion equation. In this article we study this relationship for these two classes of evolution equations with nonlocal diffusion/dispersion. This connection is especially useful, if the TW equation is not studied directly, but the existence of a TWS is proven using one of the evolution equations instead. Finally, we present three models from fluid dynamics and discuss the TWP via its link to associated reaction-diffusion equations.
Keywords
Cite
@article{arxiv.1703.05920,
title = {Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem},
author = {Franz Achleitner},
journal= {arXiv preprint arXiv:1703.05920},
year = {2023}
}