A priori bounds and global bifurcation results for frequency combs modeled by the Lugiato-Lefever equation

In nonlinear optics 2Pi-periodic solutions a 2 C2([0; 2Pi];C) of the stationary Lugiato-Lefever equation -da00 = (i-)a + jaj2a - if serve as a model for frequency combs, which are optical signals consisting of a superposition of modes with equally spaced frequencies. In accordance with experimental data we prove that nontrivial frequency combs can only be observed for special values of the forcing and detuning parameters f, . E.g., if the detuning parameter is too large then nontrivial frequency combs do not exist, cf. Theorem 2. Additionally, we show that for large ranges of parameter values nontrivial frequency combs may be found on continua which bifurcate from curves of trivial frequency combs. Our results rely on the proof of a priori bounds for the stationary Lugiato-Lefever equation as well as a detailed rigorous bifurcation analysis based on the bifurcation theorems of Crandall-Rabinowitz and Rabinowitz

Pinning in the extended Lugiato–Lefever equation

We consider a variant of the Lugiato-Lefever equation (LLE), which is a nonlinear Schrödinger equation on a one-dimensional torus with forcing and damping, to which we add a first-order derivative term with a potential $\varepsilon V(x)$. The potential breaks the translation invariance of LLE. Depending on the existence of zeroes of the effective potential $V_\text{eff}$, which is a suitably weighted and integrated version of $V$, we show that stationary solutions from $\varepsilon=0$ can be continued locally into the range $\varepsilon \ne 0$. Moreover, the extremal points of the $\varepsilon$-continued solutions are located near zeros of $V_\text{eff}$ . We therefore call this phenomenon pinning of stationary solutions. If we assume additionally that the starting stationary solution at $\varepsilon=0$ is spectrally stable with the simple zero eigenvalue due to translation invariance being the only eigenvalue on the imaginary axis, we can prove asymptotic stability or instability of its $\varepsilon$-continuation depending on the sign of $V\u27_\text{eff}$ at the zero of $V_\text{eff}$ and the sign of $\varepsilon$. The variant of the LLE arises in the description of optical frequency combs in a Kerr nonlinear ring-shaped microresonator which is pumped by two different continuous monochromatic light sources of different frequencies and different powers. Our analytical findings are illustrated by numerical simulations

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described