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- W1828618774 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. On étudie l'équation continue résonante (CR) qui a été obtenue dans [8] comme limite d'une équation de Schrödinger cubique faiblement non linéaire (ou à petites données). On commence par montrer que ce modèle est naturel puisqu'il correspond aussi à la partie résonante de l'équation d'Hermite–Schrödinger. On montre alors que la base des fonctions spéciales d'Hermite est bien adaptée à son analyse et permet de comprendre certains aspects étonnants de l'équation. On étudie certaines dynamiques particulières de l'équation sur des espaces invariants : espaces propres de l'oscillateur harmonique et de l'opérateur des rotations, espace de Bargmann–Fock. Enfin on s'intéresse aux solutions stationnaires et à leur stabilité." @default.
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- W1828618774 date "2016-01-01" @default.
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- W1828618774 title "On the continuous resonant equation for NLS. I. Deterministic analysis" @default.
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- W1828618774 doi "https://doi.org/10.1016/j.matpur.2015.10.002" @default.
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