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，研究人员已经开发出几代短脉冲、高能量光纤激光器，用于各种应用。这些激光器被配置为通过围绕环多次传播光来产生周期性的稳定脉冲。虽然不同的物理效应会改变脉冲在循环中的形状，但脉冲在每个周期（往返）都会返回相同的形状。这些激光器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，需要新的数学方法。该项目将开发理论和计算方法，以确定模拟激光系统的非线性波动方程的周期性稳定脉冲解，并分析其稳定性（在存在随机噪声和其他系统扰动的情况下的鲁棒性）。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，该项目还将支持以教学创新为重点的补充活动。 在这个项目中要研究的激光模型是基于五阶复杂的金斯堡-朗道方程的变体。经典地，定常非线性波的谱由线性化微分算子的Evans函数的零点集给出。Ginzburg-Landau方程和光纤激光器模型的周期定态解的稳定性将根据关于脉冲的线性化的单值算子的谱来表征。由于时间周期解的稳定性问题是在圆柱上而不是在真实的直线上形成的，因此Evans函数的任何推广都将涉及无限维函数空间上算子的Fredholm行列式，而不是矩阵的经典行列式。为了避免在无限维情形下用于计算埃文斯函数的微分方程的极端刚性，单值算子的点谱将与无限圆柱上的Birman-Schwinger算子的无限维Fredholm行列式的零点集相同。数值方法将被开发来计算Fredholm行列式的Birman-Schwinger运营商。然后，这些方法将被用来确定在设计参数空间的Ginzburg-Landau方程和实验光纤激光器系统的模型的周期性稳定的解决方案的稳定区域。最近，通过将一个称为Maslov指数的相关拓扑不变量应用于自伴算子的谱理论，建立了反应扩散方程定态解的一般不稳定性。一个新版本的马斯洛夫指数将被用来建立一般的稳定性结果的定期固定解决方案的金斯堡-朗道equation.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.