Coherent Structures in the Nonlinear Maxwell Equations
The primitive equations governing wave propagation in spatially varying optical fibers are the nonlinear Maxwell equations, though this process is often modeled using the nonlinear coupled mode equations (NLCME). NLCME describe the evolution of the slowly varying envelope of an appropriate carrier wave. They are known to possess solitons, which may be of use in optical transmission. In this talk, we numerically study the evolution the NLCME soliton in the primitive equations, and find them to be robust. This is highly non-trivial, as the nonlinear Maxwell equations are a non-convex hyperbolic system, requiring careful treatment of the Riemann problem. Furthermore, we consider extensions of NLCME to a system of infinitely many nonlinear coupled mode equations and present some results suggesting this new system also possesses localized solutions.