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数学进展 - 2020, Vol. 49(6): 693-712
研究论文
带有时间周期色散和时间变化损耗或增益的随机 Schr$\ddot{o}$dinger 方程
On Stochastic Schr$\ddot{o}$dinger Equation with Time-periodic Dispersion and Time-varying Loss/gain

简慧
JIAN Hui

华东交通大学理学院, 南昌, 江西, 330013
School of Science, East China Jiaotong University, Nangchang, Jiangxi, 330013, P. R. China

收稿日期: 2019-07-26
出版日期: 2020-11-17
2020, Vol. 49(6): 693-712
DOI: 10.11845/sxjz.2019086b


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摘要 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Schr$\ddot{o}$dinger 方程$\mathrm{i}\mathrm{d}u + \frac{1}{\varepsilon} m(\frac{t}{\varepsilon^{2}})\partial_{xx}u\mathrm{d}t + \nu(\frac{t}{\varepsilon})\partial_{xx}u\mathrm{d}t +\lambda|u|^{2\sigma}u \mathrm{d}t + \mathrm{i}\varepsilon a(t)u\mathrm{d}t = 0$. 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 Schr$\ddot{o}$dinger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$\varepsilon\rightarrow0$时, 在一维空间中证明了原方程解的局部渐近收敛性.
关键词 随机非线性Schr$\ddot{o}$dinger 方程随机和时间周期色散时变损耗或增益Strichartz估计非线性光学    
Abstract:This paper is concerned with a stochastic nonlinear Schr$\ddot{o}$dinger equation including deterministic time-periodic dispersion and time-varying loss/gain: $\mathrm{i}\mathrm{d}u + \frac{1}{\varepsilon} m(\frac{t}{\varepsilon^{2}})\partial_{xx}u\mathrm{d}t +\nu(\frac{t}{\varepsilon})\partial_{xx}u\mathrm{d}t +\lambda|u|^{2 \sigma}u \mathrm{d}t + \mathrm{i}\varepsilon a(t)u\mathrm{d}t = 0$, 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 Schr$\ddot{o}$dinger 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 as $\varepsilon\rightarrow0$ in one space dimension.
Key wordsstochastic nonlinear Schr$\ddot{o}$dinger equation    random and time-periodic dispersion    time-varying loss/gain    Strichartz estimate    nonlinear fibre optics
PACS:  O175.27  
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