A random dispersion Schrödinger equation with time-oscillating nonlinearity

In this paper, we consider the nonlinear Schrödinger equation with the random dispersion and time-oscillating nonlinearity,iut+1εm(tε2)∂xxu+θ(tε2)|u|2σu=0,x∈R,t>0,σ>0, where m satisfying some ergodic conditions and θ   a periodic function. We prove that the solution uεuε converges as ε→0ε→0 to the solution of the limit equationidu+∂xxu∘dβ+I(θ)|u|2σudt=0idu+∂xxu∘dβ+I(θ)|u|2σudt=0 with the initial datum u0u0 in H1(R)H1(R). And the convergence holds in the sense of distribution in C([0,T];H1(R))C([0,T];H1(R)), T<τ⁎(u0)T<τ⁎(u0) almost surely, where τ⁎(u0)τ⁎(u0) is the maximal existence time for the limit equation.