A random dispersion Schrِdinger equation with time-oscillating nonlinearity

In this paper, we consider the nonlinear Schrödinger equation with the random dispersion and time-oscillating nonlinearity, i u t + 1 ε m ( t ε 2 ) ∂ x x u + θ ( t ε 2 ) | u | 2 σ u = 0 , x ∈ R , t > 0 , σ > 0 , where m satisfying some ergodic conditions and θ a periodic function. We prove that the solution u ε converges as ε → 0 to the solution of the limit equation i d u + ∂ x x u ∘ d β + I ( θ ) | u | 2 σ u d t = 0 with the initial datum u 0 in H 1 ( R ) . And the convergence holds in the sense of distribution in C ( [ 0 , T ] ; H 1 ( R ) ) , T < τ ⁎ ( u 0 ) almost surely, where τ ⁎ ( u 0 ) is the maximal existence time for the limit equation.