Solitary waves for the power degenerate NLS -- existence and stability

TL;DR

This paper constructs solitary wave solutions for a power degenerate nonlinear Schrödinger equation using variational methods, analyzes their properties, and characterizes their spectral stability, while highlighting open conjectures.

Contribution

It introduces a variational construction of solitary waves for a degenerate NLS and provides a comprehensive stability analysis within the sharp parameter range.

Findings

Existence of solitary waves as minimizers of an inequality

Properties include positivity, smoothness, and decay

Complete spectral stability characterization

Abstract

We consider a semilinear Schr\"odinger equation, driven by the power degenerate second order differential operator $\nabla\cdot (|x|^{2a} \nabla), a\in (0,1)$. We construct the solitary waves, in the sharp range of parameters, as minimizers of the Caffarelli-Kohn-Nirenberg's inequality. Depending on the parameter $a$ and the nonlinearity, we establish a number of properties, such as positivity, smoothness (away from the origin) and almost exponential decay. Then, and as a consequence of our variational constrcution, we completely characterize the spectral stability of the said solitons. We pose some natural conjectures, which are still open -- such as the radiality of the ground states, the non-degeneracy and most importantly uniqueness.