The project concerns the development and analysis of new asymptotic methods to study short wavelength solutions of nonlinear hyperbolic partial differential equations. Short wavelength asymptotics, also known as geometric optics, is one of the most penetrating and widely applicable methods for analyzing partial differential equations. Traditionally applied to linear problems, the method has recently been employed to furnish rigorous results in the nonlinear context. This project continues the development of the method for treatment of nonlinear problems. Special emphasis is placed on analysis of solutions that model ultrashort laser pulses and on the behavior of both wave trains and pulses upon crossing focal points.
This project develops new mathematical tools to describe the propagation of waves whose form is that of short pulses. The technique, called asymptotic analysis, studies such pulses in the limit as the wavelength of the pulse becomes shorter and shorter. In such limiting situations, much can be learned about solutions of the underlying equations, which express physical laws such as Newton's second law for fluids or the laws of electromagnetism. This project develops the technique to analyze important nonlinear equations that arise in a variety of applications. Special attention is given to diffractive effects that occur when rays of geometric optics stay close together for long times, and to the effects that occur when pulses focus in a nonlinear regime. Promising preliminary results show that standard numerical techniques can be improved in both accuracy and simplicity. The results of this project will yield reliable procedures for numerical simulations of important physical systems, including lasers that produce ultrashort pulses.
