Jared Bronski : Unconditional Stability of KdV-Burgers Fronts

Posted by Liet Vo , part of the Analysis and Applied Mathematics Seminar.

At

Sept. 25, 2023, 4 p.m.

In

636 SEO

Abstract

\[ u_t + u u_x = \eta u_{xxx} + u_{xx} \qquad \lim_{x \rightarrow \mp \infty }u = \pm 1 \] Originally proposed by Whitham as a model for the propagation of tidal bores. It was shown by Bona and Schonbek that front type traveling wave solutions exist for all $\eta$, unique modulo translation, and are monotone for $|\eta|\leq \frac14$, and by Pego that such solutions are stable to small perturbations for the monotone case. We present a new stability criteria that does not require a smallness assumption on the difference between the initial data and the traveling wave, and which can be shown to hold in an open set of $\eta$ values that includes the monotone case. This condition involves the number of bound states of a certain Schr\”dinger operator constructed from the front solution. We will also discuss some rigorous numerical calculations that give intervals in $\eta$ where this spectral condition is guaranteed to hold. Joint work with Blake Barker, Vera Hur and Zhao Yang.