Solitary waves for the Hartree equation with a slowly varying potential

TL;DR

This paper analyzes the behavior of solitary waves in the Hartree equation with a slowly varying potential, showing they follow classical dynamics with controlled errors over logarithmic timescales.

Contribution

It extends existing methods to the Hartree equation, providing a detailed description of soliton evolution under slowly varying potentials and initial conditions close to solitons.

Findings

Solitons follow classical Hamiltonian dynamics with small errors.

The evolution holds up to logarithmic timescales in the inverse of the potential variation.

Extension to more general initial conditions beyond previous results.

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.