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

TL;DR

This paper investigates the existence, properties, and stability of standing wave solutions to a mixed dispersion nonlinear Schrödinger equation, providing new insights into their qualitative behavior and conditions for stability.

Contribution

The study introduces new existence results and stability analysis for standing waves in a mixed dispersion nonlinear Schrödinger equation with constrained minimization approaches.

Findings

Existence of minimizers under certain conditions.

Standing waves exhibit specific symmetry and decay properties.

Proved orbital stability of certain standing wave solutions.

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.