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We indicate that the nonlinear Schr\"odinger equation with white noisedispersion possesses stochastic symplectic and multi-symplectic structures.Based on these structures, we propose the stochastic symplectic andmulti-symplectic methods, which preserve the continuous and discrete chargeconservation laws, respectively. Moreover, we show that the proposed methodsare convergent with temporal order one in probability. Numerical experimentsare presented to verify our theoretical results.

Stochastic Symplectic and Multi-Symplectic Methods for Nonlinear Schr\"odinger Equation with White Noise Dispersion