$L^2$-contraction of Shock Waves for KdV-Burgers Equation

TL;DR

This paper proves an $L^2$-contraction property for viscous-dispersive shocks in the KdV-Burgers equation, demonstrating stability and uniform estimates under large perturbations and time-dependent shifts.

Contribution

It establishes the $L^2$-contraction for viscous-dispersive shocks in the KdV-Burgers equation, including large perturbations and time-dependent shifts, advancing stability analysis.

Findings

Proved $L^2$-contraction for viscous-dispersive shocks.

Demonstrated stability under large perturbations.

Provided uniform estimates with respect to viscosity and dispersion.

Abstract

The KdV-Burgers equation is a canonical model describing the interplay between nonlinearity, viscosity and dispersion, and it admits viscous-dispersive shocks as traveling wave solutions. In this paper, we establish an $L^2$-contraction property for viscous-dispersive shocks under arbitrarily large perturbations, up to a time-dependent shift. This yields time-asymptotic stability and uniform estimates with respect to the strengths of viscosity and dispersion. We present the proof for the monotone shocks, and introduce the companion work in [6] on the stability and structural properties of oscillatory shocks.