9704549 Forest Modulation theory and rigorous analysis of the small dispersion limit of scalar and vector nonlinear Schrodinger (NLS) equations are fundamental to data transmission in nonlinear fiber optics. For pulses in use today, so- called "non-return-to-zero" or NRZ pulses, fundamental issues consist of: how do the pulses degrade?, and, what combination of fiber properties and pulse features determine how much and where the pulse degrades? The principal investigator and Ken McLaughlin have studied these questions for two particular transoceanic fibers in use today, with answers provided to the above questions. Proposed new research involves the description of pulse propagation once degradation has begun; mathematically, these results require generalization of integrable PDE methods in two fundamental ways. First, one must provide detailed behavior of the solution after shock formation, including how fast the ripples spread and evolve with the main pulse; and second, there are no results to date for weak dispersive behavior of the coupled NLS equations which apply when there are birefringent effects, and there is no systematic construction of oscillatory solutions for coupled NLS pdes. Current data transmission questions in nonlinear fiber optics have to be answered in order to achieve a robust, faithful communication link for the various planned applications. These questions relate to: the type of fiber one uses; for a given fiber, the type of data one sends as carriers of information bits; for given fiber and data, how long can the transmission line be before unacceptable errors occur as bits become distorted. Remarkably for the applied mathematical community, each of these issues can be framed in terms of cutting-edge questions in special nonlinear partial differential equations called nearly integrable. The pulse propagation in modern nonlinear fibers is well-known to be approximated by the solution of perturbed nonlinear Schrodinger equations, special equations studied in great detail by the principal investigator for twenty years. The material properties of the fiber give coefficients in the equations, the length of the fiber dictates how long one must construct the solution, and the shape of the data that is sent along the fibers is the input for the solution. The modeling of pulse propagation and degradation involves two phases: the mathematical understanding and description of the solutions of these equations for these special types of input data; and the transfer of the mathematical insights and tools to practical understanding, identification of problems, and suggestions for remedies. The issues of how pulses degrade and how to shape pulses to minimize degradation are fundamental mathematical issues which form the core of the proposed research. The P.I. has and will continue to communicate with the technological community, consisting thus far of visits and conversations with staff scientists at Lucent Technologies and Corning, Inc.
