# American Institute of Mathematical Sciences

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

## Slow soliton interaction with delta impurities

 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, $(\xi^2 + q \, \sech^2 ( x ) )$/$2$.
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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