On the collapse of trial solutions for a damped-driven non-linear Schr\"odinger equation

TL;DR

This paper demonstrates that a physically motivated trial solution for a damped-driven 2D nonlinear Schrödinger equation with critical non-linearity does not collapse, using a stochastic Hamiltonian approach to establish global solutions.

Contribution

It introduces a novel stochastic Hamiltonian framework to prove non-collapse of trial solutions in a damped-driven nonlinear Schrödinger equation.

Findings

Trial solutions do not collapse for any admissible initial conditions.

A global solution to a singular stochastic Hamiltonian system is constructed.

The method links trial solutions with the original Schrödinger equation.

Abstract

We consider the focusing 2D non-linear Schr\"odinger equation, perturbed by a damping term, and driven by multiplicative noise. We show that a physically motivated trial solution does not collapse for any admissible initial condition although the exponent of the non-linearity is critical. Our method is based on the construction of a global solution to a singular stochastic Hamiltonian system used to connect trial solution and Schr\"odinger equation.