[AN] Rik Westdorp: Soliton Amplification in the Korteweg-de Vries Equation by Multiplicative Forcing

11 June 2024 16:00 till 17:00 - Location: EEMCS Lecture hall Chip | Add to my calendar

The Korteweg-de Vries (KdV) equation is a well-studied dispersive PDE that famously admits solitary wave solutions of various amplitudes and velocities. In this talk, we discuss the stability and dynamics of solitons in the KdV equation with small multiplicative forcing. Forcing breaks the conservative structure of the KdV equation, leading to substantial changes in energy over a long time. For small forcing, the inserted energy is almost fully absorbed by the soliton, resulting in drastically changed amplitude and velocity.

We study solutions to the forced equation using a modulation approach: decomposing solutions into a modulated soliton and an infinite-dimensional perturbation. This allows for an explicit description of amplitude/velocity dynamics and sets the stage for stability analysis. Assuming slow exponential decay of the forcing, we show that the perturbation decays at the same exponential rate in a weighted Sobolev norm centered around the soliton. The main novelty of this result is that it allows amplitude changes of arbitrary size, which significantly complicates the analysis.

This is joint work with Prof. Hermen Jan Hupkes.