9626306 Ercolani In the projects proposed here, the PI intends to concentrate on two particular areas where geometric methods are playing a fundamental role. The methods to be applied here are, first, soliton and near-integrable systems theory and, second, geometric singularity theory.The first concerns the modelling of dispersive difference schemes. The equations of nonlinear optics frequently operate in regimes where dispersive effects tend to dominate over those of viscosity. A particularly relevant example of this is the effects of numerical dispersion in modelling carrier wave shocking of femtosecond pulses. An appropriate model that can provide a benchmark for these effects is the integrable discrete nonlinear Schrodinger equations. The integrability of this model enables one to entertain a precise analysis of continuum limits for this system. The PI proposes to use the underlying Kahler geometry of this model to deduce convexity estimates which should enable one to derive a valid characterization of these limits including the appearance of modulational instabilities. These characterizations will then be compared to modulation equations derived by formal averaging as well as to numerical simulations of non-integrable discretizations of the nonlinear Schrodinger (NLS) pde. The second topic concerns pattern formation in nonlinear optics. Experiments reveal a rich variety of transverse patterns for coherent fields in a nonlinear optical cavity. The Nonlinear Optics group of the Arizona Center for Mathematical Sciences has derived mean field models for a cavity filled with an isotropic, nonlinear Kerr medium, and driven by a linearly polarised input field. The equations for this model have the form of a damped, driven coupled pair of defocussing NLS equations and have been numerically demonstrated to produce patterns and defects of the type seen experimentally. The PI and his collaborators are particularly interested to understand the formation and evolut ion of defects in pattern forming evolution equations and have developed order parameter equations which model the behaviour of modulated roll paterns in a variety of such evolution equations. The PI proposes to derive and study such modulation equations for the above mentioned mean field model. The PI has also developed a general approach, based on geometric singularity theory, for the description of singularties in systems of pde's. The goals of this project will be first, to classify generic types of defects for the nonlinear optics model and assess their temporal stability; second, to control the formation of defects to the end of either eliminating them or using them to encode information in patterns. Expectations concerning this second goal are supportied by recent sucess in controlling pattern irregularities in one dimensional models. %%% Nonlinear optics, which may be defined as the study of the interaction of intense light with matter, was born with the invention of the laser in the 1960's. In those early days such interactions required very large intensities; however, nowadays such systems can be constructed with only modest power requirements. Along with this has come the possibility of constructing large, so-called wide aperture, devices such as semiconductor lasers. Such devices will play an important role in many areas of technology such as satellite communications systems with an ultrafast switching capacity. With the introduction of these larger physical scales one expects to see the formation of patterns as well as defects in these patterns (analogous to crystalline defects) which can be modelled by the equations developed in this proposal. Understanding how such patterns arise and evolve can play an important role in efficiently operating and controlling these devices. In the second project of this proposal oscillatory solutions in a special class of interacting lattice systems will be investigated. These systems have modelling app lications which range from theoretical biology to the numerical methods used in numerically simulating optical systems. The goal is to obtain, rigorously, a continuum description, given by partial differential equations and valid on long spatial and temporal scales, of the microscopic variations in these lattice systems. Such models can help to explain how a microscopic system, such as an alpha-helix protein molecule or a coupled laser array, can behave collectively, and hence macroscopically, in order to contribute to phenomena on scales much larger than those orignally present in the microscopic system. ***
