Soliton interaction with slowly varying potentials

TL;DR

This paper analyzes how solitons in the Gross-Pitaevskii equation interact with slowly varying potentials, showing they follow classical dynamics with improved accuracy over previous results, confirmed by simulations.

Contribution

It demonstrates that solitons evolve according to an effective Hamiltonian with errors of order h^2, improving upon earlier estimates and providing rigorous and numerical validation.

Findings

Solitons follow classical dynamics up to time log(1/h)/h

Error bounds are improved to order h^2 in H^1 norm

Numerical simulations confirm theoretical predictions

Abstract

We study the Gross-Pitaevskii equation with a slowly varying smooth potential, $V(x) = W(hx)$. We show that up to time $\log(1/h)/h $ and errors of size $h^2$ in $H^1$, the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian, $ (\xi^2 + \sech^2 * V (x))/2 $. This provides an improvement ($ h \to h^2 $) compared to previous works, and is strikingly confirmed by numerical simulations.