Orbitally stable standing waves of a mixed dispersion nonlinear Schr\"odinger equation

Casteras, Jean-Baptiste; Bonheure, Denis; Santos, Ederson Moreira Dos; Nascimento, Robson

Type de document :

Pré-publication ou Document de travail

Titre :

Orbitally stable standing waves of a mixed dispersion nonlinear Schr\"odinger equation

Auteur(s) :

Casteras, Jean-Baptiste [Auteur]
Quantitative methods for stochastic models in physics [MEPHYSTO]
Département de mathématiques Université Libre de Bruxelles
Bonheure, Denis [Auteur]
Quantitative methods for stochastic models in physics [MEPHYSTO]
Santos, Ederson Moreira Dos [Auteur]
Nascimento, Robson [Auteur]
Département de mathématiques Université Libre de Bruxelles

Discipline(s) HAL :

Mathématiques [math]/Equations aux dérivées partielles [math.AP]

Résumé en anglais : [en]

We study the mixed dispersion fourth order nonlinear Schr\"odinger equation \begin{equation*} %\tag{\protect{4NLS}}\label{4nls} i \partial_t \psi -\gamma \Delta^2 \psi +\beta \Delta \psi +|\psi|^{2\sigma} \psi =0\ \text{in}\ ...
Lire la suite >We study the mixed dispersion fourth order nonlinear Schr\"odinger equation \begin{equation*} %\tag{\protect{4NLS}}\label{4nls} i \partial_t \psi -\gamma \Delta^2 \psi +\beta \Delta \psi +|\psi|^{2\sigma} \psi =0\ \text{in}\ \R \times\R^N, \end{equation*} where $\gamma,\sigma>0$ and $\beta \in \R$. We focus on standing wave solutions, namely solutions of the form $\psi (x,t)=e^{i\alpha t}u(x)$, for some $\alpha \in \R$. This ansatz yields the fourth-order elliptic equation \begin{equation*} %\tag{\protect{*}}\label{4nlsstar} \gamma \Delta^2 u -\beta \Delta u +\alpha u =|u|^{2\sigma} u. \end{equation*} We consider two associated constrained minimization problems: one with a constraint on the $L^2$-norm and the other on the $L^{2\sigma +2}$-norm. Under suitable conditions, we establish existence of minimizers and we investigate their qualitative properties, namely their sign, symmetry and decay at infinity as well as their uniqueness, nondegeneracy and orbital stability.Lire moins >

Langue :

Anglais

Collections :

Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL

Fichiers

1710.09775
Accès libre
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