On nonlinear Schrodinger type equations with nonlinear damping

TL;DR

This paper investigates nonlinear Schrödinger equations with nonlinear damping, demonstrating how damping prevents blow-up, how confinement leads to decay, and analyzing scattering behavior in the absence of external potential.

Contribution

It provides new insights into the long-term behavior of solutions to nonlinear Schrödinger equations with damping and confinement effects.

Findings

Nonlinear damping prevents finite time blow-up.

Quadratic confinement causes solutions to decay to zero over time.

Without external potential, solutions may scatter and not decay.

Abstract

We consider equations of nonlinear Schrodinger type augmented by nonlinear damping terms. We show that nonlinear damping prevents finite time blow-up in several situations, which we describe. We also prove that the presence of a quadratic confinement in all spatial directions drives the solution of our model to zero for large time. In the case without external potential we prove that the solution may not go to zero for large time due to (non-trivial) scattering.