Effective dynamics for $N$-solitons of the Gross-Pitaevskii equation

TL;DR

This paper demonstrates that the effective finite-dimensional dynamics accurately approximate the behavior of multiple solitons in the Gross-Pitaevskii equation under slowly varying external potentials, with quantifiable error bounds.

Contribution

It extends the effective dynamics approach from one soliton to multiple solitons and quantifies the approximation error in the presence of external potentials.

Findings

Effective dynamics provide a good approximation with error size h^2.

The Ehrenrest time scale log(1/h)/h is natural for unstable equilibria.

Numerical results support applicability to multiple solitons in Bose-Einstein experiments.

Abstract

We consider several solitons moving in a slowly varying external field. We show that the effective dynamics obtained by restricting the full Hamiltonian to the finite dimensional manifold of $ N$-solitons (constructed when no external field is present) provides a remarkably good approximation to the actual soliton dynamics. That is quantified as an error of size $ h^2 $ where $ h $ is the parameter describing the slowly varying nature of the potential. This also indicates that previous mathematical results of Holmer-Zworski for one soliton are optimal. For potentials with unstable equilibria the Ehrenrest time, $ \log(1/h)/h $, appears to be the natural limiting time for these effective dynamics. We also show that the results of Holmer-Perelman-Zworski for two mKdV solitons apply numerically to a larger number of interacting solitons. We illustrate the results by applying the method with the external potentials used in Bose-Einstein soliton train experiments of Strecker et. al.