Recent Results On The Modulated Cubic Nonlinear Schr\"odinger Equation On $\mathbb{T}^2$

TL;DR

This paper establishes new Strichartz estimates for the modulated cubic nonlinear Schrödinger equation on the torus, demonstrating local well-posedness and exploring effects of white noise modulation.

Contribution

It introduces novel Strichartz estimates for the modulated NLS and proves pathwise local well-posedness, including cases with white noise modulation.

Findings

Established new Strichartz estimates for the equation.

Proved local well-posedness of the modulated NLS.

Analyzed properties like mass conservation and linear flow convergence.

Abstract

New Strichartz estimates for the modulated cubic nonlinear Schr\"{o}dinger equation are proved. These Strichartz estimates allow us to show that this equation is pathwise locally well-posed. We also show that improved Strichartz estimates are available in the case where the modulation is white noise. Additionally, we comment on a few basic properties of the modulated cubic nonlinear Schr\"{o}dinger equation such as conservation of mass and convergence of its linear flow as time tends to zero.