Global solution for the stochastic nonlinear Schr\"odinger system with quadratic interaction in four dimensions

TL;DR

This paper proves the global existence of solutions for a stochastic nonlinear Schrödinger system with quadratic interactions in four dimensions, using energy estimates and ground state analysis.

Contribution

It introduces a novel approach to handle quadratic nonlinearities in a stochastic Schrödinger system at the critical dimension, leveraging energy cancellations.

Findings

Established global solutions under ground state conditions.

Developed energy derivative estimates for the stochastic system.

Addressed $L^2$-critical quadratic nonlinearities in 4D.

Abstract

We discuss the global existence of solutions to a system of stochastic Schr\"odinger equations with multiplicative noise. Our setting of the quadratic nonlinear terms in dimension 4 is $L^2$-critical. We treat the solutions under the ground state. We estimate the time derivative of the quantity of energy by using the cancellation of the cubic terms in the spatial derivative of the solution.