Slow soliton interaction with delta impurities
Justin Holmer Maciej Zworski
Journal of Modern Dynamics 2007, 1(4): 689-718 doi: 10.3934/jmd.2007.1.689
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
A remark on inverse problems for resonances
Maciej Zworski
Inverse Problems & Imaging 2007, 1(1): 225-227 doi: 10.3934/ipi.2007.1.225
Trace formulæ have been a powerful tool of inverse spectral theory on compact manifolds. We explain how the information from singularities away from zero immediately translates to the setting of resonances producing similar inverse results.
keywords: inverse problems resonances trace formula.

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