Stability and instability in scalar balance laws: fronts and periodic waves

TL;DR

This paper classifies the stability of traveling wave solutions in scalar balance laws, highlighting the critical influence of characteristic points on spectral properties and phase dynamics, and narrows down the stable wave types for a broad class of equations.

Contribution

It provides a comprehensive classification of stability and instability of scalar balance law traveling waves, emphasizing the role of characteristic points in spectral and nonlinear analysis.

Findings

Characteristic points critically influence spectra and phase dynamics.

Most entropic traveling waves are unstable, with a small subset being stable.

The analysis simplifies the variety of waves to a manageable set of stable solutions.

Abstract

We complete a full classification of non-degenerate traveling waves of scalar balance laws from the point of view of spectral and nonlinear stability/instability under (piecewise) smooth perturbations. A striking feature of our analysis is the elucidation of the prominent role of characteristic points in the determination of both the spectra of the linearized operators and the phase dynamics involved in the nonlinear large-time evolution. For a generic class of equations an upshot of our analysis is a dramatic reduction from a tremendously wide variety of entropic traveling waves to a relatively small range of stable entropic traveling waves.