[论文解读] A Study of a Nonlinear Schrödinger Equation for Optical Fibers

Domenico Felice|arXiv (Cornell University)|Dec 1, 2016

Numerical methods for differential equations被引用 6

一句话总结

本文研究了单模光纤中光传输的非线性薛定谔方程（NLSE），通过卡托方法和斯特里科夫估计，聚焦于解的局部与全局存在性。研究证明，在特定条件下，通过添加特定项所导出的修正可积NLSE与物理NLSE在L²范数下保持接近，从而可通过变换转化为标准立方NLSE，实现解析解的求解。

ABSTRACT

Non linear fiber optics concerns with the non linear optical phenomena occurring inside optical fibers. The propagation of light in single-mode fibers is governed by the one-dimensional nonlinear Schrödinger equation (NLS) in the presence of attenuation, dispersion, and non linear effects. In this NLS the role of space and time is exchanged with respect to the standard NLS as introduced in the almost entire mathematical literature. Such an exchange is far from being only formal as it enters in the interpretation of basic ideas as well-posedness and stability, as well as in the phenomenological meaning of the predictions formulated by such model. In this dissertation, the physical bases of the optical fibers are provided, and a derivation of the NLS from the Maxwell's equations is reviewed. Furthermore, problems of local nature (local existence of solutions, uniqueness) and problems of global nature (global existence) are studied by the Kato's method based on a fixed point argument and Strichartz's estimates. Moreover, tools usually employed to study global problems connected with finite-time blow up of solutions, are here used to show a closeness result between the NLS for optical fibers and an integrable NLS. Integrability is pursued by means of the Painlevé analysis that allows to describe quite a large class of integrable equations like the transformed equation from the standard NLS. Among these functions, one can be described as a standard NLS with an additional linear harmonic oscillator term. This equation is showed to be close in $L^2$-norm to the NLS for single-mode fibers. Finally, the dissertation focuses on two kinds of stationary solutions: the so-called time-homogenous, that is given by a stationary wave oscillating in space, and the soliton-like solution. Results are obtained about stability properties of them.

研究动机与目标

• 分析具有衰减、色散和非线性的单模光纤中非线性薛定谔方程解的局部与全局存在性。
• 研究在小扰动下解的稳定性，特别关注连续波和孤子型解的行为。
• 确定通过添加特定项使物理NLSE近似为可积NLSE的条件。
• 通过L²范数估计，建立物理NLSE与可积NLSE解之间的定量接近性。
• 应用变换技术，从标准立方NLSE的已知解推导可积系统的解析解。

提出的方法

• 结合卡托的不动点论证与斯特里科夫估计，证明NLSE解的局部与全局存在性。
• 采用庞特列韦分析，识别一般非自治一维薛定谔方程的可积性条件。
• 在物理NLSE中引入额外项，使其满足庞特列韦条件，从而实现可积性。
• 通过将可积非自治NLSE变换为标准立方NLSE，生成解析解。
• 在L²范数下定义距离泛函 $ \widetilde{d}_{\theta} $，用于度量物理NLSE与可积NLSE解之间的接近程度。
• 应用稳定性定理（如定理 \ref{teoremadistabilitaorbitale}），证明解在扰动下的轨道稳定性。

实验结果

研究问题

• RQ1 在何种条件下，可通过添加特定项使光纤中的物理NLSE近似为可积NLSE？
• RQ2 在推导出的条件下，物理NLSE与可积NLSE的解在$ L^2 $-范数下有多接近？
• RQ3 在存在衰减与色散的情况下，连续波与孤子型解在小扰动下的稳定性行为如何？
• RQ4 可积NLSE能否变换为标准立方NLSE？从该变换中可导出哪些解析解？
• RQ5 庞特列韦性质在识别光纤中非自治NLSE可积形式方面起什么作用？

主要发现

• 由于衰减与色散的存在，光纤中的物理NLSE不满足庞特列韦可积性条件，因此不可积。
• 通过向NLSE中添加特定项，系统变为可积，并可变换为标准立方NLSE，从而实现解析解的构造。
• 在适当的参数约束下，可积NLSE的解与物理NLSE的解在$ L^2 $-范数下保持接近，其接近程度由距离泛函 $ \widetilde{d}_{\theta} $ 量化。
• 已建立解的轨道稳定性：若初始扰动足够小（在$ \widetilde{d}_{\theta} $意义下），则解在所有$ z \geq 0 $范围内均保持在参考解轨道附近。
• 该接近性结果通过标准NLSE的有限时间爆破分析技术得到证明，表明该近似具有鲁棒性。
• 该方法可导出可积系统的解析孤子型与连续波解，且可映射回物理系统。