Soliton Amplification in the Korteweg-de Vries Equation by Multiplicative Forcing

We study the stability and dynamics of solitons in the Korteweg-de Vries (KdV) equation with small multiplicative forcing. Forcing breaks the conservative structure of the KdV equation, leading to substantial changes in energy over long times. We show that, for small forcing, the inserted energy is almost fully absorbed by the soliton, resulting in a drastically changed amplitude and velocity. We decompose the solution to the forced equation into a modulated soliton and an infinite dimensional perturbation. 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.