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

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.