Inverse Scattering Transform and non-decaying solutions of completely integrable nonlinear PDE's

University Of Alaska Fairbanks Campus, Fairbanks AK

Investigators

Abstract

This project continues to investigate an extension of the inverse scattering transform (IST) method of solving integrable nonlinear evolution PDEs (partial differential equations) to handle initial data in a larger class. In other words, the project focuses on soliton theory for initial profiles that are much broader than rapidly decaying or periodic. It is well-known that slow decay at infinity may lead to new phenomena. For instance, certain smooth but slowly decaying initial data may turn into rough or even blow-up revealing a very complicated relation between local and global behaviors. Standard techniques of PDEs or numerical analysis are ineffective to tackle such issues. The IST is much better suited to study such phenomena as it combines both global and local properties of initial data, linearizes the problem, and provides accurate asymptotic behavior. Although the spectrum of the underlying differential operators (e.g. Schrödinger in the case of the Korteweg-de Vries (KdV) equation) is much more complicated than for the classical IST, it can be suitably expressed in terms of the Titchmarsh-Weyl m-function which is well-defined for virtually any reasonable initial profile. The main thrust of this project is a study of the IST in this setting. In particular, the effect of different spectral components of the differential operator in the Lax pair on the solution of the corresponding nonlinear PDE will be investigated. The inverse scattering transform was first discovered in the 60s for the KdV equation of shallow-water waves. Soon after, it was found for many other important nonlinear PDEs and now 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. The importance of extending the range of validity of IST, which is the principal aim of the project, is recognized by both mathematicians and physicists. The results are expected to be of a very applied nature and could be employed for the study of wave propagation on different backgrounds (including noisy), tidal waves, certain meteorological phenomena, understanding freak waves or any other applied problems where initial data do not approach zero at infinity, and in such diverse disciplines as hydrodynamics, telecommunication, atmospheric sciences, nonlinear optics, plasma, astrophysics, etc. Given their remarkable pedigree, diversity of mathematics involved, and richness of applications, the topics of the project provide a great educational experience through research for the undergraduate and graduate students involved.