Solutions for fourth order anisotropic nonlinear Schr\"odinger equations in $\R^2$

TL;DR

This paper investigates solutions to a fourth order anisotropic nonlinear Schrödinger equation in two dimensions, establishing well-posedness, blowup, and properties of standing waves, highlighting differences from isotropic cases.

Contribution

It provides new results on existence, stability, and decay of solutions for an anisotropic nonlinear Schrödinger equation, extending previous isotropic analyses.

Findings

Proved local and global well-posedness and blowup conditions.

Established existence and stability properties of standing waves.

Demonstrated differences from isotropic Schrödinger equations.

Abstract

In this paper, we consider solutions to the following fourth order anisotropic nonlinear Schr\"odinger equation in $\R \times \R^2$, $$ \left\{ \begin{aligned} &\textnormal{i}\partial_t\psi+\partial_{xx} \psi-\partial_{yyyy} \psi +|\psi|^{p-2} \psi=0, \\ &\psi(0)=\psi_0 \in H^{1,2}(\R^2), \end{aligned} \right. $$ where $p>2$. First we prove the local/global well-posedness and blowup of solutions to the Cauchy problem for the anisotropic nonlinear Schr\"odinger equation. Then we establish the existence, axial symmetry, exponential decay and orbital stability/instability of standing waves to the anisotropic nonlinear Schr\"odinger equation. The pictures are considerably different from the ones for the isotropic nonlinear Schr\"odinger equations. The results are easily extendable to the higher dimensional case.