Scattering theory for CMV operators and applications to completely integrable systems

Completely integrable wave equations, also known as soliton equations, are an important topic in physics used to explain many physical phenomena. More than 200 years after their discovery by John Scott Russell, solitons are still a highly active area in both physics and mathematics producing new and fascinating phenomena. One of the main tools from quantum mechanics for solving completely integrable nonlinear wave equations is known as the inverse scattering transform, first discovered and applied for the Korteweg-de Vries equation by Clifford Gardner, John Greene, Martin Kruskal, and Robert Miura in 1967. This method involves the solution of an underlying spectral problem, namely the scattering problem for the associated Lax operator. The aim of this project is to start the investigation of the scattering problem for CMV operators (Maria J. Cantero, Leandro Moral, and Luis Velazquez, 2003), which have recently been shown to be the Lax operators for the discrete analogue of the nonlinear Schrödinger equation, the Ablowitz-Ladik system. This will allow me to solve the initial value problem for the Ablowitz-Ladik equation and eventually for the whole hierarchy of differential- difference equations associated with the Ablowitz-Ladik system. This hierarchy will be investigated within the framework of this project as well, my approach will be via a zero-curvature equation. My analysis will add to the methods of solving spatially discrete evolution equations like the Toda hierarchy. The nonlinear Schrödinger equation describes the evolution of small amplitude, slowly varying wave packets in nonlinear media. For example, it has been derived in such diverse fields as deep water waves, plasma physics, nonlinear optical fibers, and so on.