On the standing waves of the Schrödinger equation with concentrated nonlinearity

We study the concentrated NLS on R n , with power non-linearities, driven by the fractional Laplacian, ( - Δ ) s , s > 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 ( R n ) . Next, we provide a complete...

Full description

Saved in:
Bibliographic Details
Published in:Analysis and mathematical physics 2021-09, Vol.11 (3), Article 136
Main Authors: Ramadan, Abba, Stefanov, Atanas G.
Format: Article
Language:English
Subjects:
Analysis
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Nonlinearity
Schrodinger equation
Solitary waves
Standing waves
Citations: Items that this one cites
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We study the concentrated NLS on R n , with power non-linearities, driven by the fractional Laplacian, ( - Δ ) s , s > 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 ( R n ) . 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 ( R n ) ↪ L ∞ ( R n ) , which provides an alternative calculation and justification of the sharp constants in these inequalities.
ISSN:1664-2368
1664-235X
DOI:10.1007/s13324-021-00565-6