Existence and Stability of Traveling Waves for an Integro-differential Equation for Slow Erosion

TL;DR

This paper investigates an integro-differential equation modeling slow erosion in granular flow, establishing the existence, uniqueness, and local stability of traveling wave solutions that differ from those in standard conservation laws.

Contribution

It introduces the first analysis of traveling waves for this specific integro-differential equation, demonstrating their existence, uniqueness, and stability.

Findings

Existence of unique traveling wave solutions

Traveling waves connect profiles with equilibrium slopes at infinity

Traveling wave profiles are locally stable and attract solutions with monotone initial data

Abstract

We study an integro-differential equation that describes the slow erosion of granular flow. The equation is a first order non-linear conservation law where the flux function includes an integral term. We show that there exist unique traveling wave solutions that connect profiles with equilibrium slope at $\pm\infty$. Such traveling waves take very different forms from those in standard conservation laws. Furthermore, we prove that the traveling wave profiles are locally stable, i.e., solutions with monotone initial data approaches the traveling waves asymptotically as $t\to+\infty$.