Stochastic nonlinear Schrödinger equation on an upper-right quarter plane with Dirichlet random boundary

Abstract

In this paper, we study the nonhomogeneous stochastic initial-boundary value problem for the nonlinear Schrödinger equation on an upper-right quarter plane with random Dirichlet boundary conditions. The main novelty of this work is a convenient framework for the analysis of such equations excited by the Wiener additive noise on the boundary. Our approach allows us to show the local existence and uniqueness of solutions in the space H2. The basic properties of the solutions such as the continuity and the boundary-layer behavior are also studied using the Itô calculus. Despite several technical difficulties, we believe that the approach developed in this paper can be applied to the case of a large class of noise including fractional Wiener space time white noise, homogeneous noise, and Levy noise.