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arXiv · 1703.05920

Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem

Franz Achleitner
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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.

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Franz Achleitner. 2017-03-17. Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem. https://doi.org/10.1007/978-3-319-66839-0_2

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