Asymptotic behavior of solutions to a dissipative nonlinear Schr\"odinger equation with time dependent harmonic potentials

TL;DR

This paper investigates the long-term behavior of solutions to a dissipative nonlinear Schrödinger equation with a time-dependent harmonic potential, identifying conditions under which solutions decay or persist.

Contribution

It analyzes the critical conditions affecting the decay of solutions, linking nonlinear power and potential decay to solution behavior.

Findings

Decay of solutions depends on nonlinear power and potential decay rate.

Identifies a critical threshold for solution decay or persistence.

Provides conditions for asymptotic behavior of dissipative Schrödinger solutions.

Abstract

We consider the Cauchy problem of a dissipative nonlinear Schr\"odinger equation with a time dependent harmonic potential. We find a critical situation that the $L^2$-norm of dissipative solutions decays or not and which is decided by a nonlinear power and time decay order of harmonic potential.