An inhomogeneous, $L^2$ critical, nonlinear Schr\"odinger equation

TL;DR

This paper investigates an inhomogeneous, $L^2$ critical nonlinear Schrödinger equation, establishing conditions for global solutions and demonstrating the strong instability of standing waves through finite-time blow-up solutions.

Contribution

It provides the sharp $L^2$ norm condition for global existence and proves strong instability of standing waves via self-similar blow-up solutions.

Findings

Sharp $L^2$ norm condition for global existence

Construction of self-similar solutions blowing up in finite time

Proof of strong instability of standing waves

Abstract

An inhomogeneous nonlinear Schr\"odinger equation is considered, that is invariant under $L^2$ scaling. The sharp condition for global existence of $H^1$ solutions is established, involving the $L^2$ norm of the ground state of the stationary equation. Strong instability of standing waves is proved by constructing self-similar solutions blowing up in finite time.