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

James Fennell|arXiv (Cornell University)|Apr 22, 2018

Cold Atom Physics and Bose-Einstein Condensates4 citations

TL;DR

This paper studies resonant Hamiltonian systems derived from the one-dimensional nonlinear Schrödinger equation with harmonic trapping, proving they approximate the full NLS dynamics in the small data, long-time regime. It establishes global wellposedness, invariance under Fourier transform and harmonic oscillator flow, and identifies an infinite family of explicit Hermite-function stationary solutions.

ABSTRACT

We study two resonant Hamiltonian systems on the phase space $L^2(\\mathbb{R} \ ightarrow \\mathbb{C})$: 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.

Motivation & Objective

• To analyze the dynamics of resonant Hamiltonian systems arising from the quintic and cubic one-dimensional nonlinear Schrödinger equation with harmonic trapping.
• To prove that these resonant systems accurately approximate the full NLS dynamics in the small data, long-time regime.
• To investigate intrinsic dynamical features of the resonant systems, including symmetries, wellposedness, and stationary solutions.
• To establish a robust wellposedness theory for the resonant systems in $L^2$ and higher Sobolev spaces.
• To identify and characterize an infinite family of orthogonal, explicit stationary wave solutions in the form of Hermite functions.

Proposed method

• Derives the continuous resonant (CR) equation for the quintic and cubic NLS with harmonic trapping via a normal form reduction in the small data limit.
• Applies techniques from dispersive PDEs and space-time resonance theory to justify the approximation of full NLS dynamics by the resonant systems.
• Establishes global wellposedness in $L^2$ and all higher $L^2$ Sobolev spaces using energy estimates and symmetry-based arguments.
• Identifies invariance of the resonant systems under the Fourier transform and the linear Schrödinger flow with harmonic trapping.
• Constructs an infinite family of explicit stationary solutions using Hermite functions, which are orthogonal in $L^2$.
• Analyzes the Hamiltonian structure of the resonant systems to reveal their underlying symmetries and conservation laws.

Experimental results

Research questions

• RQ1 How accurately do the resonant Hamiltonian systems approximate the dynamics of the full quintic and cubic NLS with harmonic trapping in the small data regime?
• RQ2 What are the intrinsic dynamical features of the resonant systems, such as symmetries and conservation laws?
• RQ3 Does the resonant system admit a robust wellposedness theory, including global existence and uniqueness?
• RQ4 Are there explicit, orthogonal stationary solutions in the resonant system, and what is their structure?
• RQ5 To what extent are the resonant systems invariant under transformations like the Fourier transform and harmonic oscillator evolution?

Key findings

• The resonant systems provide a valid long-time approximation to the full quintic and cubic NLS with harmonic trapping under small initial data.
• The resonant systems are globally wellposed in $L^2$ and all higher $L^2$ Sobolev spaces, ensuring long-time existence and uniqueness of solutions.
• The systems exhibit non-trivial symmetries, including invariance under the Fourier transform and the linear Schrödinger evolution with harmonic trapping.
• An infinite family of orthogonal, explicit stationary wave solutions exists, parameterized by Hermite functions.
• The stationary solutions are eigenfunctions of the resonant Hamiltonian system and form a complete orthogonal basis in $L^2(\mathbb{R}; \mathbb{C})$.
• The resonant systems are shown to be Hamiltonian with a well-defined Poisson structure, enabling rigorous analysis of their dynamics.