Long-Time Dynamics of Variable Coefficient mKdV Solitary Waves

Details Long-Time Dynamics of Variable Coefficient mKdV Solitary Waves Long-Time Dynamics of Variable Coefficient mKdV Solitary Waves

2005-03-08

arxiv.org/abs/math-ph/0503016v1

J. Math. Phys. 47, 072703 (2006)

Physics

Mathematical Physics

19 pages

Scientific paper

10.1063/1.2217809

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.

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