Stochastic Quantization of Laser Propagation Models

Abstract

This paper identifies certain interesting mathematical problems of stochastic quantization type in the modeling of Laser propagation through turbulent media. In some of the typical physical contexts the problem reduces to stochastic Schrodinger equation with space-time white noise of Gaussian, Poisson and Levy type. We identify their mathematical resolution via stochastic quantization. Nonlinear phenomena such as Kerr effect can be modeled by stochastic nonlinear Schrodinger equation in the focusing case with space-time white noise. A treatment of stochastic transport equation, the Korteweg-de Vries Equation as well as a number of other nonlinear wave equations with space-time white noise is also given. Main technique is the S-transform (we will actually use closely related Hermite transform) which converts the stochastic partial differential equation with space time white noise to a deterministic partial differential equation defined on the Hida-Kondratiev white noise distribution space. We then utlize the inverse S-transform/Hermite transform known as the characterization theorem combined with the infinite dimensional implicit function theorem for analytic maps to establish local existence and uniqueness theorems for pathwise solutions of these class of problems. The particular focus of this paper on singular white noise distributions is motivated by practical situations where the refractive index fluctuations in propagation medium in space and time are intense due to turbulence, ionospheric plasma turbulence, marine-layer fluctuations, etc. Since a large class of partial differential equations that arise in nonlinear wave propagation have polynomial type nonlinearities, white noise distribution theory is an effective tool in studying these problems subject to different types of white noises.