On periodic boundary solutions for cylindrical and spherical KdV-Burgers equations

TL;DR

This paper investigates the evolution of periodic boundary perturbations in cylindrical and spherical KdV-Burgers equations, revealing the formation of shock chains and explicit head shock asymptotics in dissipative and dispersive media.

Contribution

It provides a detailed analysis of the development of periodic boundary solutions and derives explicit asymptotics for head shocks in cylindrical and spherical geometries.

Findings

Formation of shock chains with decreasing amplitude

Explicit asymptotic for head shock height and velocity

Dependence of shock characteristics on spatial dimension and boundary conditions

Abstract

For the KdV-Burgers equations for cylindrical and spherical waves the development of a regular profile starting from an equilibrium under a periodic perturbation at the boundary is studied. The equation describes a medium which is both dissipative and dispersive. For an appropriate combination of dispersion and dissipation the asymptotic profile looks like a periodical chain of shock fronts with a decreasing amplitude (sawtooth waves). The development of such a profile is preceded by a head shock of a constant height and equal velocity which depends on spatial dimension as well as on integral characteristics of boundary condition; an explicit asymptotic for this head shock is found.