Undular bore theory for the modified Korteweg-de Vries-Burgers equation

TL;DR

This paper develops an analytical theory for undular bores in the modified Korteweg-de Vries-Burgers equation, incorporating small viscosity effects, and validates it against numerical solutions, showing stabilization of wave structures.

Contribution

It introduces a Whitham modulation approach that includes viscosity as a perturbation for the mKdV-Burgers equation, providing new insights into wave stabilization.

Findings

Small viscosity stabilizes cnoidal bores over time

Analytical results agree well with numerical simulations

Main characteristics of undular bores are derived analytically

Abstract

We consider nonlinear wave structures described by the modified Korteweg-de Vries equation with taking into account a small Burgers viscosity for the case of step-like initial conditions. The Whitham modulation equations are derived which include the small viscosity as a perturbation. It is shown that for long enough time of evolution this small perturbation leads to stabilization of cnoidal bores and their main characteristics are obtained. Applicability conditions of this approach are discussed. Analytical theory is compared with numerical solutions and good agreement is found.