Finite time extinction for nonlinear Schrodinger equation in 1D and 2D

TL;DR

This paper investigates conditions under which solutions to the nonlinear Schrödinger equation in 1D and 2D vanish in finite time, highlighting the role of damping and initial data size.

Contribution

It demonstrates finite time extinction for nonlinear Schrödinger equations with damping on compact manifolds and the whole space in low dimensions, depending on damping and initial data.

Findings

Sublinear damping causes finite time extinction in 1D.

Small initial mass leads to extinction in 2D.

Extra damping prevents blow-up in the studied models.

Abstract

We consider a nonlinear Schrodinger equation with power nonlinearity, either on a compact manifold without boundary, or on the whole space in the presence of harmonic confinement, in space dimension one and two. Up to introducing an extra superlinear damping to prevent finite time blow up, we show that the presence of a sublinear damping always leads to finite time extinction of the solution in 1D, and that the same phenomenon is present in the case of small mass initial data in 2D.