Orbital instability of periodic waves for scalar viscous balance laws

TL;DR

This paper proves that spectral instability of periodic traveling waves in scalar viscous balance laws leads to their nonlinear orbital instability, using a framework based on well-posedness and bifurcation analysis.

Contribution

It establishes a general instability criterion linking spectral and orbital instability for a broad class of scalar viscous balance laws and applies it to specific wave families.

Findings

Spectral instability implies orbital instability for the waves studied.

The criterion applies to small amplitude waves near Hopf bifurcation.

The criterion also applies to large period waves approaching traveling pulses.

Abstract

The purpose of this paper is to prove that, for a large class of nonlinear evolution equations known as scalar viscous balance laws, the spectral (linear) instability condition of periodic traveling wave solutions implies their orbital (nonlinear) instability in appropriate periodic Sobolev spaces. The analysis is based on the well-posedness theory, the smoothness of the data-solution map, and an abstract result of instability of equilibria under nonlinear iterations. The resulting instability criterion is applied to two families of periodic waves. The first family consists of small amplitude waves with finite fundamental period which emerge from a local Hopf bifurcation around a critical value of the velocity. The second family comprises arbitrarily large period waves which arise from a homoclinic (global) bifurcation and tend to a limiting traveling pulse when their fundamental period tends to infinity. In the case of both families, the criterion is applied to conclude their orbital instability under the flow of the nonlinear viscous balance law in periodic Sobolev spaces with same period as the fundamental period of the wave.