Long-Time Dynamics of Variable Coefficient mKdV Solitary Waves

TL;DR

This paper analyzes the long-time behavior of solutions to a variable coefficient mKdV equation, showing that solutions near a solitary wave maintain a wave form with evolving parameters and small fluctuations over extended periods.

Contribution

It introduces a framework for understanding the long-time dynamics of variable coefficient mKdV solitary waves, including the evolution of wave parameters influenced by slowly varying coefficients.

Findings

Solutions stay close to a solitary wave over long times

Wave center and scale follow a specific dynamical law

Small fluctuations remain controlled in H^1 norm

Abstract

We study the Korteweg-de Vries-type equation dt u=-dx(dx^2 u+f(u)-B(t,x)u), where B is a small and bounded, slowly varying function and f is a nonlinearity. Many variable coefficient KdV-type equations can be rescaled into this equation. We study the long time behaviour of solutions with initial conditions close to a stable, B=0 solitary wave. We prove that for long time intervals, such solutions have the form of the solitary wave, whose centre and scale evolve according to a certain dynamical law involving the function B(t,x), plus an H^1-small fluctuation.