Solitary waves for the Hartree equation with a slowly varying potential

Details Solitary waves for the Hartree equation with a slowly varying potential Solitary waves for the Hartree equation with a slowly varying potential

2009-04-06

arxiv.org/abs/0904.0834v1

Mathematics

Analysis of PDEs

28 pages

Scientific paper

We study the Hartree equation with a slowly varying smooth potential, $V(x) = W(hx)$, and with an initial condition which is $\epsilon \le \sqrt h$ away in $H^1$ from a soliton. We show that up to time $|\log h|/h$ and errors of size $\epsilon + h^2$ in $H^1$, the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian. This result is based on methods of Holmer-Zworski, who prove a similar theorem for the Gross-Pitaevskii equation, and on spectral estimates for the linearized Hartree operator recently obtained by Lenzmann. We also provide an extension of the result of Holmer-Zworski to more general inital conditions.

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