Convergence rate toward shock wave under periodic perturbation for generalized Korteweg-de Vries-Burgers equation

TL;DR

This paper studies how solutions to the generalized Korteweg-de Vries-Burgers equation with small periodic perturbations converge to a viscous shock wave, establishing exponential decay rates and the influence of oscillations.

Contribution

It demonstrates the asymptotic convergence of solutions to a viscous shock wave under periodic perturbations and quantifies the decay rate and shift caused by oscillations.

Findings

Solutions tend to a shifted viscous shock wave over time.

Exponential decay rate towards the shock wave is established.

Periodic oscillations influence the shock wave's position.

Abstract

In this paper, a viscous shock wave under space-periodic perturbation of generalized Korteweg-de Vries-Burgers equation is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover the exponential time decay rate toward the viscous shock wave is also obtained for some certain perturbations.