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

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

Document type :

Pré-publication ou Document de travail

Title :

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

Author(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

HAL domain(s) :

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

English abstract : [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}\ ...
Show more >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.Show less >

Language :

Anglais

Collections :

Laboratoire Paul Painlevé - UMR 8524

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1710.09775
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