On the standing waves of the Schroedinger equation with concentrated nonlinearity

TL;DR

This paper constructs and classifies the spectral stability of solitary waves for a fractional nonlinear Schrödinger equation with concentrated nonlinearity, showing their orbital stability and linking soliton profiles to sharp Sobolev embedding constants.

Contribution

It explicitly constructs solitary waves for the fractional NLS with concentrated nonlinearity and classifies their spectral stability, establishing their orbital stability and connection to Sobolev inequalities.

Findings

Explicit solitary wave solutions in the energy space

Complete spectral stability classification

Orbital stability of spectrally stable waves

Abstract

We study the concentrated NLS on ${\mathbf R^n}$, with power non-linearities, driven by the fractional Laplacian, $(-\Delta)^s, s>\frac{n}{2}$. We construct the solitary waves explicitly, in an optimal range of the parameters, so that they belong to the natural energy space $H^s$. Next, we provide a complete classification of their spectral stability. Finally, we show that the waves are non-degenerate and consequently orbitally stable, whenever they are spectrally stable. Incidentally, our construction shows that the soliton profiles for the concentrated NLS are in fact exact minimizers of the Sobolev embedding $H^s({\mathbf R^n})\hookrightarrow L^\infty({\mathbf R^n})$, which provides an alternative calculation and justification of the sharp constants in these inequalities.