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

自20世纪90年代孤子激光器问世以来，研究人员已经开发了几代短脉冲、高能光纤激光器，用于各种应用。这些激光器被配置成通过在环路上多次传播光来产生周期性的固定脉冲。虽然不同的物理效应会改变脉冲在环路中的形状，但脉冲在每个周期（往返）都会恢复到相同的形状。对这些激光器进行建模的一个重大挑战是，从一代到下一代，脉冲的呼吸量急剧增加，需要新的数学方法。该项目将发展理论和计算方法，以确定非线性波动方程建模激光系统的周期性平稳脉冲解，并分析其稳定性（随机噪声和其他系统扰动存在的鲁棒性）。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器，以及用于高度精确测量时间和频率的频率梳，并应用于地理定位系统、时间和频率标准、天文仪器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，该项目将支持以教学创新为重点的补充活动。本项目研究的激光模型是基于三次五次复金兹堡-朗道方程的变体。经典地，平稳非线性波的谱由线性化微分算子的埃文斯函数的零集给出。金兹堡-朗道方程周期平稳解和光纤激光器模型的稳定性将由脉冲线性化的单算子谱来表征。由于时间周期解的稳定性问题是在柱体上而不是在实线上表述的，所以埃文斯函数的任何推广都将涉及无穷维函数空间上算子的Fredholm行列式，而不是矩阵的经典行列式。为了避免在这种无限维情况下用于计算Evans函数的微分方程的极端刚度，一元算子的点谱将被标识为无限柱面上Birman-Schwinger算子的无限维Fredholm行列式的零集。将发展数值方法来计算此类Birman-Schwinger算子的Fredholm行列式。这些方法将用于确定金兹堡-朗道方程周期性平稳解和实验光纤激光系统模型在设计参数空间中的稳定区域。最近，在自伴随算子的谱理论中应用了一个相关的拓扑不变量马斯洛夫指数，建立了反应扩散方程平稳解的一般不稳定性。一个新版本的马斯洛夫指数将被用来建立一般稳定性结果的周期平稳解的金兹堡-朗道方程。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.