Existence and stability of standing waves for nonlinear fractional Schr\"odinger equation with logarithmic nonlinearity

TL;DR

This paper investigates the existence, stability, and properties of standing wave solutions for a nonlinear fractional Schrödinger equation with a logarithmic nonlinearity, using variational and compactness methods.

Contribution

It establishes the existence and uniqueness of global solutions, constructs ground states as minimizers, and proves their stability under the dynamics.

Findings

Existence of ground state solutions as minimizers on the Nehari manifold

Global well-posedness of the Cauchy problem for the equation

Stability of the set of minimizers under the evolution

Abstract

In this paper we consider the nonlinear fractional logarithmic Schr\"{o}dinger equation. By using a compactness method, we construct a unique global solution of the associated Cauchy problem in a suitable functional framework. We also prove the existence of ground states as minimizers of the action on the Nehari manifold. Finally, we prove that the set of minimizers is a stable set for the initial value problem, that is, a solution whose initial data is near the set will remain near it for all time.