Solitary waves for the Hartree equation with a slowly varying potential

Abstract

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.

28 pages

References in corpus (4)

• Solitary Wave Dynamics in an External Potential
• Uniqueness of Ground States for Pseudo-Relativistic Hartree Equations
• Long time motion of NLS solitary waves in a confining potential
• Solitary wave dynamics in time-dependent potentials

Cited by in corpus (2)

• On the soliton dynamics under slowly varying medium for Nonlinear Schrödinger equations
• Phase-driven interaction of widely separated nonlinear Schrödinger solitons