Extinction of multiple shocks in the modular Burgers equation

TL;DR

This paper investigates the behavior of multiple shock waves in the modular Burgers' equation, revealing finite-time coalescence of interfaces through energy estimates and numerical simulations, and establishing a scaling law for this extinction process.

Contribution

It introduces the analysis of multiple shock interactions in the modular Burgers' equation and derives a scaling law for shock coalescence, extending previous single-shock stability results.

Findings

Multiple shocks coalesce in finite time

A scaling law for shock extinction is formulated

Numerical simulations support the theoretical predictions

Abstract

We consider multiple shock waves in the Burgers' equation with a modular advection term. It was previously shown that the modular Burgers' equation admits a traveling viscous shock with a single interface, which is stable against smooth and exponentially localized perturbations. In contrast, we suggest in the present work with the help of energy estimates and numerical simulations that the evolution of shock waves with multiple interfaces leads to finite-time coalescence of two consecutive interfaces. We formulate a precise scaling law of the finite-time extinction supported by the interface equations and by numerical simulations.