Dafermos Regularization of a Modified KdV-Burgers Equation

Abstract

This project involves Dafermos regularization of a partial differential equation of order higher than 2. The modified Korteweg de Vries-Burgers equation is

u_T + f(u)_X = alpha u_XX +beta u_XXX,

where the flux is f(u) = u^3, alpha> 0, and beta is nonzero. We show the existence of Riemann-Dafermos solutions near a given Riemann solution composed of shock waves using geometric singular perturbation theory. When beta > 0, there is a possibility that the Riemann solution is composed of two shock waves as opposed to one. In addition, we use linearization to study the stability of the Riemann-Dafermos solutions.