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Collaborative Research: Stability and Instability of Periodically Stationary Nonlinear Waves with Applications to Fiber Lasers

合作研究：周期性平稳非线性波的稳定性和不稳定性及其在光纤激光器中的应用

基本信息

批准号：
2106203
负责人：
John Zweck
金额：
$ 24.09万
依托单位：
University of Texas at Dallas
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-09-01 至 2025-02-28
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2106203&HistoricalAwards=false
关键词：
Collaborative Research Stability Instability Periodically

项目摘要

Since the introduction of the soliton laser in the 1990's researchers have developed several generations of short pulse, high energy fiber lasers for a variety of applications. These lasers are configured to produce periodically stationary pulses by propagating light many times around a loop. Although different physical effects change the shape of the pulse as it traverses the loop, the pulse returns to the same shape once each period (round trip). A significant challenge for the modeling of these lasers is that from one generation to the next there has been a dramatic increase in the amount by which the pulse breathes, necessitating novel mathematical approaches. This project will develop theoretical and computational methods to determine periodically stationary pulse solutions of nonlinear wave equations modeling laser systems and to analyze their stability (robustness in the presence of random noise and other system perturbations). The project will provide computational tools to aid in the design of high energy lasers for medical applications, and of frequency combs for highly accurate measurements of time and frequency, with applications to geo-location systems, time and frequency standards, the calibration of astronomical instruments, and trace gas sensing. The project will provide broad training in applied mathematics for doctoral students and mentoring for junior faculty. In addition, the project will support complementary activity focused on pedagogical innovations. The laser models to be studied in this project are based on variants of the cubic-quintic complex Ginzburg-Landau equation. Classically, the spectrum of a stationary nonlinear wave is given by the zero set of the Evans function of the linearized differential operator. The stability of periodically stationary solutions of the Ginzburg-Landau equation and of models of fiber lasers will be characterized in terms of the spectrum of the monodromy operator of the linearization about the pulse. Since the stability problem for time-periodic solutions is formulated on a cylinder, rather than on the real line, any generalization of the Evans function will involve Fredholm determinants of operators on infinite-dimensional function spaces rather than classical determinants of matrices. To avoid the extreme stiffness of the differential equations used to compute the Evans function in this infinite dimensional context, the point spectrum of the monodromy operator will be identified with the zero set of an infinite-dimensional Fredholm determinant of a Birman-Schwinger operator on an infinite cylinder. Numerical methods will be developed to compute Fredholm determinants of such Birman-Schwinger operators. These methods will then be employed to determine stability regions in design parameter space for periodically stationary solutions of the Ginzburg-Landau equation and of models of experimental fiber laser systems. The generic instability of stationary solutions of reaction diffusion equations has recently been established by applying a related topological invariant called the Maslov index to the spectral theory of self-adjoint operators. A novel version of the Maslov index will be used to establish general stability results for periodically stationary solutions of the Ginzburg-Landau equation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

自从20世纪90年代孤子激光器问世以来，S的研究人员已经开发出了几代短脉冲、高能光纤激光器，用于各种应用。这些激光器被配置为通过围绕环路多次传播光来产生周期性稳定的脉冲。尽管不同的物理效应会在脉冲穿过环路时改变脉冲的形状，但脉冲在每个周期(往返)一次返回到相同的形状。对这些激光进行建模的一个重大挑战是，从一代到下一代，脉冲呼吸的数量急剧增加，这就需要新的数学方法。该项目将开发理论和计算方法来确定非线性波动方程模拟激光系统的周期平稳脉冲解，并分析其稳定性(在存在随机噪声和其他系统扰动时的稳健性)。该项目将提供计算工具，以帮助设计用于医疗应用的高能激光，以及用于高精度时间和频率测量的频率梳，并将其应用于地理定位系统、时间和频率标准、天文仪器的校准和痕量气体传感。该项目将为博士生提供广泛的应用数学培训，并为初级教师提供指导。此外，该项目将支持侧重于教学创新的补充活动。本项目中要研究的激光模型是基于三次-五次复数Ginzburg-Landau方程的变体。经典地，定常的非线性波谱是由线性化的微分算子的Evans函数的零集给出的。Ginzburg-Landau方程和光纤激光器模型的周期定常解的稳定性将用脉冲线性化的单调算符的谱来表征。由于时间周期解的稳定性问题是在柱面上而不是在实直线上描述的，所以Evans函数的任何推广都将涉及无限维函数空间上的算子的Fredholm型行列式，而不是经典的矩阵行列式。为了避免用于计算无限维环境中的Evans函数的微分方程式的极端僵硬，单调算子的点谱将与无限圆柱上的Bman-Schwinger算子的无穷维Fredholm型行列式的零集相一致。将发展数值方法来计算这类Bman-Schwinger算子的Fredholm型行列式。然后，这些方法将被用于确定Ginzburg-Landau方程和实验光纤激光系统模型的周期定常解在设计参数空间中的稳定域。最近，通过将一个称为Maslov指数的相关拓扑不变量应用到自伴算子的谱理论中，建立了反应扩散方程定常解的一般不稳定性。一个新版本的Maslov指数将被用来为金兹堡-朗道方程的周期平稳解建立一般稳定性结果。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。