### 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 language | English (US) |
---|---|

Pages (from-to) | 131-163 |

Number of pages | 33 |

Journal | Journal des Mathematiques Pures et Appliquees |

Volume | 105 |

Issue number | 1 |

DOIs | |

State | Published - 2016 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Applied Mathematics
- Mathematics(all)

### Cite this

*Journal des Mathematiques Pures et Appliquees*,

*105*(1), 131-163. https://doi.org/10.1016/j.matpur.2015.10.002

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

Research output: Contribution to journal › Article

*Journal des Mathematiques Pures et Appliquees*, vol. 105, no. 1, pp. 131-163. https://doi.org/10.1016/j.matpur.2015.10.002

}

TY - JOUR

T1 - On the continuous resonant equation for NLS. I. Deterministic analysis

AU - Germain, Pierre

AU - Hani, Zaher

AU - Thomann, Laurent

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

KW - Harmonic oscillator

KW - Lowest landau level

KW - Nonlinear schrödinger equation

KW - Resonant equation

KW - Stationary solutions

UR - http://www.scopus.com/inward/record.url?scp=84960897712&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84960897712&partnerID=8YFLogxK

U2 - 10.1016/j.matpur.2015.10.002

DO - 10.1016/j.matpur.2015.10.002

M3 - Article

AN - SCOPUS:84960897712

VL - 105

SP - 131

EP - 163

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

SN - 0021-7824

IS - 1

ER -