Long-Time Dynamics of KdV Solitary Waves over a Variable Bottom

2004/11/17 by S. I. Dejak, I. M. Sigal, Dejak, S. I. +1 · 2 citations
Mathematics · Physics and Astronomy · #35Q35 #35Q53 #37K40 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP #msc:35Q35 #msc:35Q53 #msc:37K40

paper · pdf · doi:10.48550/arxiv.math-ph/0411059

33 pages

arxiv created 2004/11/17 · arxiv updated 2009/12/01

Abstract

We study the variable bottom generalized Korteweg-de Vries (bKdV) equation dt u=-dx(dx² u+f(u)-b(t,x)u), where f is a nonlinearity and b is a small, bounded and slowly varying function related to the varying depth of a channel of water. Many variable coefficient KdV-type equations, including the variable coefficient, variable bottom KdV equation, can be rescaled into the bKdV. 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 H¹-small fluctuation.

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• Dynamics of dissipative periodic optical waves in an external potential
• Soliton,breathers,positons and rogue waves for the vector complex modified Korteweg-de Vries equation

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