Analysis and Applied Mathematics Seminar

Jared Bronski
University of Illinois Urbana-Champaign
Unconditional Stability of KdV-Burgers Fronts
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.
Monday September 25, 2023 at 4:00 PM in 636 SEO
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