Georgia Institute of Technology College of Sciences

We develop a mathematical model for ultra-short pulse propagation in nonlinear metamaterials characterized by a weak Kerr-type nonlinearity in their dielectric response. The fundamental equation in the model is the short-pulse equation (SPE) which will be derived in frequency band gaps. We use a multi-scale ansatz to relate the SPE to the nonlinear Schroedinger equation, thereby characterizing the change of width of the pulse from the ultra short regime to the classical slow varying envelope approximation. We will discuss families of solutions of the SPE in characteristic coordinates, as well as discussing the global wellposedness of generalizations of the model that describe uni- and bi-directional nonlinear waves.