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

## TL;DR

This paper explores the relationship between nonlocal reaction-diffusion and Korteweg-de Vries-Burgers equations through traveling wave problems, highlighting their interconnectedness and applications in fluid dynamics.

## Contribution

It establishes a link between nonlocal reaction-diffusion and KdVB equations via traveling wave problems, enabling analysis through related models.

## Key findings

- Traveling wave solutions exist for both classes of equations.
- The connection simplifies the study of traveling waves in complex models.
- Applications demonstrated in fluid dynamics models.

## 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.