This project investigates an extension of the inverse scattering transform (IST) method of solution for integrable nonlinear partial differential equations to handle initial data in a larger class. The scattering data used in connection with short-range potentials will be replaced by data involving the Titchmarsh-Weyl m-function to develop an inverse spectral transform that extends the range of validity of the method to initial conditions that include long-range and oscillatory functions. The work will establish a properly regularized Marchenko equation that permits reconstruction of the potential. Computational simulations to guide the analysis are planned.
The main goal of the proposed research is to extend methods for explicit solution of certain nonlinear evolution equations to accommodate more realistic classes of initial data. This will be achieved by linking together two remarkable theories, Soliton Theory and Titchmarsh-Weyl Theory. Soliton theory originated in the striking discovery of an unexpected link between the Korteweg-de Vries equation in nonlinear hydrodynamics and quantum scattering theory. It is regarded as a fundamental breakthrough in mathematics, connecting different branches of pure mathematics and theoretical physics, with numerous applications ranging from hydrodynamics and nonlinear optics to astrophysics and elementary particle theory. Titchmarsh-Weyl theory was developed in the connection with the Sturm-Liouville problem, which has become a cornerstone of the spectral analysis of ordinary differential operators. This project is devoted to developing an approach to soliton theory that takes advantage of Titchmarsh-Weyl theory. The work will provide improved mathematical understanding of integrable nonlinear evolution systems, which can stimulate new developments in the study of nonlinear waves in hydrodynamics, fiber optics communication, and plasma theory.
