ASYMPTOTIC STABILITY OF VISCOUS SHOCKS IN THE MODULAR BURGERS EQUATION

Uyen Le (1) , Dmitry E Pelinovsky (2) , Pascal Poullet (3)

1 McMaster University [Hamilton, Ontario]
2 Department of Mathematics and Statistics, McMaster University, Hamilton, Canada
3 LAMIA - Laboratoire de Mathématiques Informatique et Applications

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.

Mots clés: modular Burgers equation; traveling fronts; asymptotic stability

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Uyen Le, Dmitry E Pelinovsky, Pascal Poullet. ASYMPTOTIC STABILITY OF VISCOUS SHOCKS IN THE MODULAR BURGERS EQUATION. Nonlinearity, In press. ⟨hal-03041514⟩