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Journal of Modern Dynamics (JMD)
 

Slow soliton interaction with delta impurities

Pages: 689 - 718, Issue 4, October 2007      doi:10.3934/jmd.2007.1.689

 
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Justin Holmer - Mathematics Department, University of California, Evans Hall, Berkeley, CA 94720, United States (email)
Maciej Zworski - Mathematics Department, University of California, Evans Hall, Berkeley, CA 94720, United States (email)

Abstract: 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, effective Hamiltonian, Gross-Pitaevskii equation, nonlinear Schrödinger equation, Diracmass potential, semiclassical.
Mathematics Subject Classification:  Primary: 35Q55.

Received: March 2007;      Revised: June 2007;      Available Online: July 2007.