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October  2007, 1(4): 689-718. doi: 10.3934/jmd.2007.1.689

Slow soliton interaction with delta impurities

Justin Holmer 1, and  Maciej Zworski 1,

1. 

Mathematics Department, University of California, Evans Hall, Berkeley, CA 94720, United States

Received  March 2007 Revised  June 2007 Published  July 2007

We study the Gross--Pitaevskii equation with a delta function potential, $ q \delta_0 $, where $ |q| $ is small and analyze the solutions for which the initial condition is a soliton with initial velocity $ v_0 $. We show that up to time $ (|q| + v_0^2 )^{-1/2} \log$($1$/$|q|$) the bulk of the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian, $ (\xi^2 + q \, \sech^2 ( x ) )$/$2$.
Keywords: soliton, semiclassical., Gross-Pitaevskii equation, effective Hamiltonian, Diracmass potential, nonlinear Schrödinger equation.
Mathematics Subject Classification: Primary: 35Q5.
Citation: Justin Holmer, Maciej Zworski. Slow soliton interaction with delta impurities. Journal of Modern Dynamics, 2007, 1 (4) : 689-718. doi: 10.3934/jmd.2007.1.689
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