Asymptotic stability of viscous shocks in the modular Burgers equation

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