带有时间周期色散和时间变化损耗或增益的随机Schrodinger方程

On Stochastic Schrodinger Equation with Time-periodic Dispersion and Time-varying Loss/gain

本文考虑了一类在非线性光学中出现的带有时间周期色散和时间变化损耗或增益的随机非线性Schrodinger方程idu+1/εm(t/ε2)■xxudt+ν(t/ε)+■xxudt+λ|u|^2σudt+iεa(t)udt=0.我们首先修正了de Bouard和Debussche的文献[J.Funct.Anal.,2010,259(5):1300-1321]中建立的Strichartz型估计,然后利用它们证明了含有白噪声色散的随机Schrodinger方程的局部适定性.该随机方程是原方程的极限模型.最后,当参数ε→0时,在一维空间中证明了原方程解的局部渐近收敛性.

This paper is concerned with a stochastic nonlinear Schrodinger equation including deterministic time-periodic dispersion and time-varying loss/gain:id u+1/εm(t/ε2)■xxudt+ν(t/ε)■xxudt+λ|u|^2σudt+iεa(t)udt=0,which appears in nonlinear fibre optics.We first modify the Strichartz-type estimates established by de Bouard and Debussche[J.Funct.Anal.,2010,259(5):1300-1321],and then apply them to prove the local well-posedness for a stochastic Schrodinger equation with white noise dispersion,which is the limit model of the original equation.Finally,we demonstrate the local asymptotic convergence of solution for the original equation asε→0 in one space dimension.