This project concerns the development of semiclassical asymptotic techniques for initial-value problems and initial/boundary-value problems of nonlinear wave propagation in the situation that the equation of interest is completely integrable, possessing a Lax pair representation and all of the corresponding mathematical framework. Specific problems to be addressed include (i) the weakly dispersive asymptotics of the intermediate long-wave equation and its degeneration to the Benjamin-Ono equation, (ii) mixed initial/boundary value problems for the semiclassical defocusing nonlinear Schroedinger equation on the half-line, (iii) initial-value problems for the semiclassically scaled 3-wave interaction equations, (iv) transsonic initial-value problems for the semiclassical modified nonlinear Schroedinger equation, and (v) large-degree asymptotics for rational solutions of Painleve equations. Solving these problems will require the development of new tools of asymptotic analysis for nonlocal and higher-dimensional Riemann-Hilbert analytic factorization problems.
This project involves models for the propagation of waves of various sorts. In particular, the models are capable of making predictions about the behavior of physical systems ranging in diversity from optical fibers carrying high-intensity pulses of light to the ocean, where contaminants can become trapped at the boundary between layers of fluids of differing density. Our aim is to provide practical methods for accurately approximating the solutions of these models so that their predictions can be compared with experiments and ultimately used in place of expensive or invasive experiments themselves. In particular, we note that in the wake of environmental disasters like the Deepwater Horizon oil leak of 2010, it becomes increasingly important to have accurate and easy-to-use methods for predicting the location of contaminants in the fluid column of the ocean.
