Dynamics of KdV solitons in the presence of a slowly varying potential

2010/01/11 by Justin Holmer, Holmer, Justin
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP

paper · pdf · doi:10.48550/arxiv.1001.1583

arxiv created 2010/12/31 · arxiv updated 2011/01/04

Abstract

We study the dynamics of solitons as solutions to the perturbed KdV (pKdV) equation ∂_t u = -∂_x (∂_x² u + 3u²-bu), where b(x,t) = b₀(hx,ht), h≪ 1 is a slowly varying, but not small, potential. We option an explicit description of the trajectory of the soliton parameters of scale and position on the dynamically relevant time scale δh^-1log h^-1, together with an estimate on the error of size h^1/2. In addition to the Lyapunov analysis commonly applied to these problems, we use a local virial estimate due to Martel-Merle (2005). The results are supported by numerics. The proof does not rely on the inverse scattering machinery and is expected to carry through for the L² subcritical gKdV-p equation, 1<p<5. The case of p=3, the modified Korteweg-de Vries (mKdV) equation, is structurally simpler and more precise results can be obtained by the method of Holmer-Zworski (2007).

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