Asymptotic stability of viscous shocks in the modular Burgers equation

TL;DR

This paper proves the asymptotic stability of viscous shocks in the modular Burgers equation, handling both odd and general perturbations through reduction to linear equations and weighted energy methods, supported by numerical illustrations.

Contribution

It establishes the first rigorous proof of asymptotic stability for viscous shocks in the modular Burgers equation, including general perturbations.

Findings

Viscous shocks are asymptotically stable under odd perturbations.

Stability under general perturbations is proven using weighted energy estimates.

Numerical simulations support the theoretical stability results.

Abstract

Dynamics of viscous shocks is considered in the modular Burgers equation, where the time evolution becomes complicated due to singularities produced by the modular nonlinearity. We prove that the viscous shocks are asymptotically stable under odd and general perturbations. For the odd perturbations, the proof relies on the reduction of the modular Burgers equation to a linear diffusion equation on a half-line. For the general perturbations, the proof is developed by converting the time-evolution problem to a system of linear equations coupled with a nonlinear equation for the interface position. Exponential weights in space are imposed on the initial data of general perturbations in order to gain the asymptotic decay of perturbations in time. We give numerical illustrations of asymptotic stability of the viscous shocks under general perturbations.