Vol. 248, No. 1, 2010

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Solitary waves for the Hartree equation with a slowly varying potential

Kiril Datchev and Ivan Ventura

Vol. 248 (2010), No. 1, 63–90
Abstract

We study the Hartree equation with a slowly varying smooth potential, V (x) = W(hx), and with an initial condition that is 𝜀 √h-- away in H1 from a soliton. We show that up to time |log h|∕h and errors of size 𝜀 + h2 in H1, the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian. This result is based on methods of Holmer and 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 and Zworski to more general initial conditions.

Keywords
solitons, nonlinear Schrödinger equation, effective dynamics, Hartree equation
Mathematical Subject Classification 2000
Primary: 35Q55
Secondary: 35Q51
Milestones
Received: 27 July 2009
Accepted: 5 May 2010
Published: 1 October 2010
Authors
Kiril Datchev
Mathematics Department
University of California
Berkeley, CA 94720
United States
Ivan Ventura
Mathematics Department
University of California
Berkeley, CA 94720
United States