A random dispersion Schrödinger equation with time-oscillating nonlinearity
A random dispersion Schrödinger equation with time-oscillating nonlinearity
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DOI：
10.1016/j.jmaa.2014.04.020
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发表时间：
2014-10
影响因子：
1.3
通讯作者：
D. Fang;Linzi Zhang;Ting Zhang
中科院分区：
文献类型：
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作者：
D. Fang;Linzi Zhang;Ting Zhang
本文考虑具有随机色散和时间振荡非线性的非线性Schrödinger方程，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，其中m满足一些遍历条件，θ是一个周期函数。我们证明了解u ε收敛于极限方程i d u+∂x x u°d β+ i (θ)| u| 2 σ u d d t= 0的解，初始基准点为H 1 (R) u 0。在C ([0, T]; h1 (R))， T< τ (u 0)的分布意义下收敛性基本成立，其中τ (u 0)是极限方程的最大存在时间。
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.