New insights into the solutions of a class of anisotropic nonlinear Schr\"{o}dinger equations on the plane

TL;DR

This paper investigates the existence, blow-up, and traveling wave solutions of an anisotropic nonlinear Schrödinger equation with fractional derivatives, covering subcritical to supercritical regimes and highlighting unique behaviors at the critical fractional order.

Contribution

It provides new existence and non-existence results for normalized and traveling wave solutions of the anisotropic fractional Schrödinger equation without symmetry assumptions.

Findings

Existence of normalized solutions in various regimes.

Conditions for solution blow-up.

Existence and decay of boosted traveling waves for s ≥ 1/2.

Abstract

In this paper, we study the following anisotropic nonlinear Schr\"odinger equation on the plane, \[ \begin{cases} {\rm i}\partial_t \Phi+\partial_{xx} \Phi -D_y^{2s} \Phi +|\Phi|^{p-2}\Phi=0,&\quad (t,x,y)\in\mathbb{R} \times \mathbb{R}^2, \Phi(x,y,0)=\Phi_0(x,y),&\quad (x,y)\in\mathbb{R}^2, \end{cases} \] where $D_y^{2s}=\left(-\partial_{yy}\right)^s$ denotes the fractional Laplacian with $0<s<1$ and $2<p<\frac{2(1+s)}{1-s}$. We first study the existence of normalized solutions to this equation in the subcritical, critical, and supercritical cases. To this aim, regularity results and a Pohozaev type identity are necessary. Then, we determine the conditions under which the solutions blow up. Furthermore, we demonstrate the existence of boosted traveling waves when $s\geq1/2$ and their decay at infinity. Additionally, for the delicate case $s=1/2$, we provide a non-existence result of boosted traveling waves and we establish that there is no scattering for small data. Finally, we also study normalized boosted travelling waves in the mass subcritical case. Due to the nature of the equation, we do not impose any radial symmetry on the initial data or on the solutions.