Soliton interaction with slowly varying potentials

Justin Holmer; Maciej Zworski

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arXiv · 0709.0478

Soliton interaction with slowly varying potentials

Justin Holmer·Maciej Zworski

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Abstract

We study the Gross-Pitaevskii equation with a slowly varying smooth potential, $V(x) = W(hx)$. We show that up to time $\log(1/h)/h $ and errors of size $h^2$ in $H^1$, the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian, $ (ξ^2 + \sech^2 * V (x))/2 $. This provides an improvement ($ h \to h^2 $) compared to previous works, and is strikingly confirmed by numerical simulations.

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Justin Holmer, Maciej Zworski. 2007-09-24. Soliton interaction with slowly varying potentials. https://arxiv.org/abs/0709.0478

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