Abstract

I will discuss the global nonlinear asymptotic stability of the traveling front solutions to the Korteweg-de Vries—Burgers equation, and for a general class of dispersive-dissipative perturbations of the Burgers equation. Earlier works made strong use of the monotonicity of the front, for relatively weak dispersion effects. We instead exploit the modulation of the translation parameter of the front solution, establishing a new stability criterion that a certain Schrodinger operator in one dimension has exactly one negative eigenvalue, so that a rank-one perturbation of the operator can be made positive definite. Counting the number of bound states of the Schrodinger equation, we find a sufficient condition in terms of a Bargman-type functional, related to the area between the front and the corresponding ideal shock. We analytically verify that our stability criterion is met for an open set including all monotone fronts. Our numerical experiments, revealing more stable fronts, suggest a computer-assisted proof. Joint with Blake Barker, Jared Bronski, and Zhao Yang.