\funding Opportunities in Nsf" Opportunities for Funding in Nsf Will Be Discussed

\Nonlinear Schrr odinger Equations, Wavelength Divison Multiplexing and Four Wave Mixing" In the limit of large frequency spacing the multisoliton solutions of the nonlinear Schrr odinger equation (NLS), which models the evolution of nonlinear waves in optical bers, are investigated. In Fourier space the individual pulses are found to always be widely separated, and hence can be easily, measured. In this limit, an answer to the long outstanding question of how to distinguish amongst the various solitons when they are undergoing interaction is found. It is shown that due to multisoliton soliton interactions of four wave type, signiicant contributions in the Fourier spectra are found at precisely the frequency locations predicted by asymptotic analysis. This can include signiicant deviations at a central frequency, assuming suitable combinations of soliton frequencies are initially present. The deviations should be large enough for experimental observation. It is well known that the nonlinear Schrr odinger equation has a local semi-discretization (a diieren-tial-diierence analog) which can be solved exactly. It turns out that the vector nonlinear Schrr odinger equations, which play an important role in nonlinear optics, also have a local semi-discrete analog which is solvable. These discrete equations and their soliton solutions will be presented. Finally, if time permits, the mean square timing shift which is due to the noise resulting from multisoliton soliton collisions in wavelength division NLS systems will be discussed. We show that Inverse Scattering Theory deenes a homeomorphism of L 2 onto itself. This transform is nonlinear but analogous to the Fourier transform. We use it do develop an Inverse Scattering Theory of distributions and use it to solve the KdV equation with distributions as initial data. The upshot is some old and some new (?) solutions.