Institute for Mathematics and Its Applications
Philip Holmes, Princeton University
I will discuss recent joint work with J. Nathan Kutz, Alex Mielke, and others in which variants and derivatives of the nonlinear Schroedinger equation (NLS), describing the propagation of light in optical fibers, are studied. In the first, we study a model of a broadband laser in the form of an NLS with non-local terms describing the averaged properties of a quantum mirror, the saturable Bragg reflector. We prove existence-uniqueness results and study the bifurcations and stability of certain "chirped soliton" solutions, comparing them with experimental results from W. Knox's group at Lucent Technologies. In the second, we derive a planar mapping approximating variations in amplitude and phase of a pulse propagating in a lossless optical fiber with periodically varying dispersion. The map's behavior agrees well with simulations of the periodically switched NLS due to S. Evangelides. We analyse the bifurcations of fixed points and global dynamics with a view to describing pulse modulation properties.