Longtime behavior of coupled wave equations for semiconductor lasers

TL;DR

This paper analyzes a coupled wave equation model for semiconductor lasers, proving solution existence, smooth dependence on parameters, and constructing a low-dimensional invariant manifold to study long-term dynamics.

Contribution

It introduces a novel analysis of a hyperbolic PDE-ODE coupled system, establishing solution properties and invariant manifolds for long-term behavior analysis.

Findings

Existence and smooth dependence of solutions established.

Construction of a low-dimensional invariant manifold.

Flow on the manifold described by bifurcation-amenable ODEs.

Abstract

Coupled wave equations are popular tool for investigating longitudinal dynamical effects in semiconductor lasers, for example, sensitivity to delayed optical feedback. We study a model that consists of a hyperbolic linear system of partial differential equations with one spatial dimension, which is nonlinearly coupled with a slow subsystem of ordinary differential equations. We first prove the basic statements about the existence of solutions of the initial-boundary-value problem and their smooth dependence on initial values and parameters. Hence, the model constitutes a smooth infinite-dimensional dynamical system. Then we exploit the particular slow-fast structure of the system to construct a low-dimensional attracting invariant manifold for certain parameter constellations. The flow on this invariant manifold is described by a system of ordinary differential equations that is accessible to classical bifurcation theory and numerical tools such as AUTO.