Stochastic soliton and rogue wave solutions of the nonlinear Schrödinger equation with white Gaussian noise

In this letter, we propose a novel integrable nonlinear Schr & ouml;dinger equation and its Lax pair, influenced by Brownian motion and white Gaussian noise. We aim to construct and solve new integrable systems affected by the white Gaussian noise. Utilising the classical and generalised Darboux transformations, the stochastic soliton solutions and the stochastic rogue wave solutions of this novel integrable nonlinear Schr & ouml;dinger equation are obtained and expressed in determinant form. Studies of stochastic soliton and rogue wave solutions of the NLS equation are essential for complex physical and mathematical phenomena where nonlinear interactions and randomness play crucial roles.