Fermi-Pasta-Ulam phenomena and persistent breathers in the harmonic trap

TL;DR

This paper investigates long-term dynamics of the 2D nonlinear Schrödinger equation with a harmonic trap, revealing Fermi-Pasta-Ulam-like recurrence phenomena and persistent breathers due to resonant mode interactions.

Contribution

It introduces a resonant Hamiltonian framework to analyze FPU-like recurrences and long-lived breathers in trapped nonlinear Schrödinger systems, connecting these phenomena to physical applications.

Findings

Identification of FPU-like recurrence phenomena in the NLS with harmonic trap

Discovery of long-lived breather states with stable mode spectra

Insights into resonant mode interactions governing long-term dynamics

Abstract

We consider the long-term weakly nonlinear evolution governed by the two-dimensional nonlinear Schr\"{o}dinger (NLS) equation with an isotropic harmonic oscillator potential. The dynamics in this regime is dominated by resonant interactions between quartets of linear normal modes, accurately captured by the corresponding resonant Hamiltonian system. In the framework of this system, we identify Fermi-Pasta-Ulam-like recurrence phenomena, whereby the normal-mode spectrum passes in close proximity of the initial configuration, and two-mode states with time-independent mode amplitude spectra that translate into long-lived breathers of the original NLS equation. We comment on possible implications of these findings for nonlinear optics and matter-wave dynamics in Bose-Einstein condensates.