Instability of an integrable nonlocal NLS

TL;DR

This paper investigates the global dynamics of an integrable nonlocal NLS, revealing that it admits finite time blow-up solutions with small initial data and that solitons are unstable, contrasting with the local NLS behavior.

Contribution

It demonstrates the existence of small-data blow-up solutions and soliton instability in a nonlocal NLS, contrasting with the local NLS case.

Findings

Finite time blow-up solutions exist with small initial data.

Solitons are unstable by blow-up in the nonlocal NLS.

Contrasts with the behavior of local cubic focusing NLS.

Abstract

In this note we discuss the global dynamics of an integrable nonlocal NLS on $\mathbb{R}$, which has been the object of recent investigation by integrable systems methods. We prove two results which are in striking contrast with the case of the local cubic focusing NLS on $\mathbb{R}$. First, finite time blow-up solutions exist with arbitrarily small initial data in $H^s(\mathbb{R})$, for any $s\geqslant0$. On the other hand, the solitons of the local NLS, which are also solutions of the nonlocal equation, are unstable by blow-up for the latter.