This article explores the exact wave solutions of the coupled one-dimensional stochastic non-linear Schrödinger equation with Kerr-law nonlinearity, incorporating multiplicative white noise in the Itô interpretation. The generalized projective Riccati equations (GPREs) scheme is employed to obtain the obtained solutions. The study presents a range of soliton solutions, including dark, bright, singular solitons, in addition to solutions for rational and periodic functions. Finally, the MATLAB software was used to explain the numerical simulation of some soliton solutions................ Keywords: ..........Generalized projective Riccati equations scheme; Non-linear stochastic Schrödinger equation; Exact solutions; Soliton solutions; White noise.