Numerical simulation of Lugiato-Lefever equation for Kerr combs generation in Fabry-Perot resonators

Lugiato-Lefever equation (LLE) is a nonlinear Schrödinger equation with damping, detuning and driving terms, introduced as a model for Kerr combs generation in ring-shape resonators and more recently, in the form of a variant, in Fabry-Perot (FP) resonators. The aim of this paper is to present some numerical methods that complement each other to solve the LLE in its general form both in the dynamic and in the steady state regimes. We also provide some mathematical properties of the LLE likely to help the understanding and interpretation of the numerical simulation results.