Stability of Nonlinear Waves and Spectral-Scattering Problems Using Krein Signature and Pontryagin Spaces

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

The goal of this project is to study existence and stability of nonlinear waves and spectrum of scattering problems arising in integrable systems. Particular problems studied are: extension of the Evans function technique for detection of unstable eigenvalues to three-dimensional and non-local problems in Bose-Einstein condensates; stability of nonlinear waves in strongly coupled Korteweg-de Vries (KdV) equations in a regime when local perturbation techniques fail; and application of Krein signature and Pontryagin spaces to the theory of integrable systems, particularly the study of spectral problems originated in the inverse scattering theory associated with the nonlinear Schrodinger and the Sine-Gordon equations. Mathematical analysis of existence and stability of nonlinear waves in mathematical models in nonlinear optics, condensed matter physics, or chemical processes in human brain, has far-reaching consequences for applications as analytical results often guide future experiments in physics, chemistry, or medicine. A typical feature which distinguishes the field of nonlinear waves from many other fields of pure mathematics is a case study, many problems demonstrate very similar features but they do not rely on any common theory. In the recent years a particular theme appeared recurrently in two close fields - nonlinear waves and integrable systems. It is useful to consider a problem in a space which allows existence of states with 'negative' energy. These states then may lead to instabilities of coherent structures. For stable structures these negative energy states must be either not active or completely eliminated. The aim of the project is to gain more insight into this topic by solving interesting particular applied problems and apply it to build and simplify the general theory.