Resonant Hamiltonian systems associated to the one-dimensional nonlinear Schr\"odinger equation with harmonic trapping

TL;DR

This paper investigates resonant Hamiltonian systems related to the 1D nonlinear Schrödinger equation with harmonic trapping, demonstrating their approximation of NLS dynamics and revealing rich structural and dynamical properties including symmetries, well-posedness, and explicit stationary solutions.

Contribution

It introduces and analyzes specific resonant Hamiltonian systems, showing their approximation to NLS dynamics and uncovering their symmetry, well-posedness, and explicit stationary solutions.

Findings

Resonant systems approximate NLS dynamics in the small data regime.

Resonant equations exhibit symmetries like Fourier invariance and harmonic trapping flow.

The systems have global well-posedness and explicit stationary solutions.

Abstract

We study two resonant Hamiltonian systems on the phase space $L^2(\mathbb{R} \rightarrow \mathbb{C})$: the quintic one-dimensional continuous resonant equation, and a cubic resonant system that has appeared in the literature as a modified scattering limit for an NLS equation with cigar shaped trap. We prove that these systems approximate the dynamics of the quintic and cubic one-dimensional NLS with harmonic trapping in the small data regime on long times scales. We then pursue a thorough study of the dynamics of the resonant systems themselves. Our central finding is that these resonant equations fit into a larger class of Hamiltonian systems that have many striking dynamical features: non-trivial symmetries such as invariance under the Fourier transform and the flow of the linear Scr\"odinger equation with harmonic trapping, a robust wellposedness theory, including global wellposedness in $L^2$ and all higher $L^2$ Sobolev spaces, and an infinite family of orthogonal, explicit stationary wave solutions in the form of the Hermite functions.