Nonlinear waves

The areas of study during the period of this grant include: A class of physically significant singular nonlinear integro-differential equations; the Riemann-Hilbert boundary value problem and the Inverse Scattering Transform method; Direct linearizing methods for certain nonlinear wave equations and associated nonlinear ordinary differential equations; Focusing instabilities and the propagation of water waves; Ordinary differential equations of Painelve type; and numerical schemes for nonlinear wave equations solvable by the Inverse Scattering Transform method. Research has also been carried out into the following nonlinear systems: Stimulated Raman Scattering and two-photon absorption; Nonlinear resonances and spherical solutions; Nonlinear interactions of Whistler Waves; Exact three-dimensional solutions of the three-wave resonant interaction; Discrete versions of the nonlinear Schrodinger Equation; and Thermal effects on solution propagation.