A Stochastic Parametrically-Forced NLS Equation

Abstract

In this thesis, a variation on the nonlinear Schrödinger (NLS) equation with multiplicative noise is studied. In particular, we consider a stochastic version of the parametrically-forced nonlinear Schrödinger equation (PFNLS), which models the effect of linear loss and the compensation thereof by phase-sensitive amplification in pulse propagation through optical fibers. We establish global existence and uniqueness of mild solutions for initial data in L2(R) and H1(R).
The proof is an adaptation of a fixed-point argument employed by de Bouard and Debussche [Comm. Math. Phys., 205:161-181, 1999] for the nonlinear Schrödinger equation with multiplicative noise. The fixed-point argument relies on space-time estimates on the semigroup generated by the linear parametrically-forced Schrödinger operator. We prove these so-called Strichartz estimates, originally proven for the Schrödinger operator, using Fourier methods. A key difference between the Schrödinger operator and its parametrically-forced version is that the latter is not self-adjoint. We overcome this complication by establishing fixed-time estimates on the semigroup and its adjoint, based on their Fourier representations. We also briefly discuss possible future research in the direction of stability of solitary standing wave solutions of the PFNLS equation under the influence of multiplicative noise. Using informal calculations, we demonstrate an approach to track the displacement of a soliton due to small stochastic forcing.