On the continuous resonant equation for NLS. I. Deterministic analysis

Pierre Germain, Zaher Hani, Laurent Thomann

Research output: Contribution to journalArticle

Abstract

We study the continuous resonant (CR) equation which was derived in [8] as the large-box limit of the cubic nonlinear Schrödinger equation in the small nonlinearity (or small data) regime. We first show that the system arises in another natural way, as it also corresponds to the resonant cubic Hermite-Schrödinger equation (NLS with harmonic trapping). We then establish that the basis of special Hermite functions is well suited to its analysis, and uncover more of the striking structure of the equation. We study in particular the dynamics on a few invariant subspaces: eigenspaces of the harmonic oscillator, of the rotation operator, and the Bargmann-Fock space. We focus on stationary waves and their stability.

Original languageEnglish (US)
Pages (from-to)131-163
Number of pages33
JournalJournal des Mathematiques Pures et Appliquees
Volume105
Issue number1
DOIs
StatePublished - 2016

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Nonlinear equations
NLS Equation
Hermite Functions
Cubic equation
Eigenspace
Fock Space
Special Functions
Invariant Subspace
Hermite
Trapping
Harmonic Oscillator
Nonlinear Equations
Harmonic
Nonlinearity
Operator

Keywords

  • Harmonic oscillator
  • Lowest landau level
  • Nonlinear schrödinger equation
  • Resonant equation
  • Stationary solutions

ASJC Scopus subject areas

  • Applied Mathematics
  • Mathematics(all)

Cite this

On the continuous resonant equation for NLS. I. Deterministic analysis. / Germain, Pierre; Hani, Zaher; Thomann, Laurent.

In: Journal des Mathematiques Pures et Appliquees, Vol. 105, No. 1, 2016, p. 131-163.

Research output: Contribution to journalArticle

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