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

1 MEPHYSTO - Quantitative methods for stochastic models in physics
LPP - Laboratoire Paul Painlevé - UMR 8524, ULB - Université Libre de Bruxelles [Bruxelles], Inria Lille - Nord Europe
Abstract : 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.
Document type :
Preprints, Working Papers, ...

https://hal.archives-ouvertes.fr/hal-01665515
Contributor : Jean-Baptiste Casteras <>
Submitted on : Saturday, December 16, 2017 - 7:39:08 AM
Last modification on : Tuesday, July 3, 2018 - 11:34:43 AM

### Citation

Jean-Baptiste Casteras, Denis Bonheure, Ederson Moreira Dos Santos, Robson Nascimento. Orbitally stable standing waves of a mixed dispersion nonlinear Schr\"odinger equation. 2017. ⟨hal-01665515⟩

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