A mathematical analysis of Kerr frequency combs modeled by the Lugiato-Lefever equation

Résumé

The Lugiato-Lefever equation is a nonlinear Schrödinger-type equation with damping, detuning and driving, derived in nonlinear optics by Lugiato and Lefever (1987). While intensively studied in the physics literature, there are relatively few rigorous mathematical studies of this equation. Of particular interest for the physical problem is the formation and the dynamical behavior of Kerr frequency combs (optical signals consisting of a super-position of modes with equally spaced frequencies). The underlying mathematical questions concern the existence and the stability of certain particular steady solutions of the Lugiato-Lefever equation. In this talk, I'll focus on periodic steady waves for which I'll show how tools from bifurcation theory can be used to study their existence and stability.