This project will focus on the analysis of the properties of solutions for both linear and nonlinear Schrodinger equations and on the analysis of ground states and evolution in the models of Skyrme and Faddeev. The principal investigator will continue his study of the exact and approximate fundamental solutions for time-dependent Schrodinger equations on noncompact and compact manifolds, with special emphasis on estimates relevant for nonlinear problems. For the models of Skyrme and Faddeev, he will study the existence of ground states and their regularity. Also, he will study the Cauchy problem for these models. The PI will extend the techniques developed recently in connection with wave maps and Schrodinger maps to these more complicated sigma-models. In addition, he will initiate computational analysis of the ground states and the Cauchy problem in the case when the base manifold is compact.
The Schrodinger equation is the basic equation of quantum physics. The properties of its solutions for different physical systems and environments are of great importance to our understanding of quantum phenomena. On the other hand, to describe and better understand elementary particles, physicists design various mathematical models, or field theories. Of special interest are the models that use subtle topological invariants. The models of Skyrme and Faddeev are examples of such models. Their analysis poses challenging mathematical problems on the intersection of the theory of partial differential equations and geometry/topology and may bring new physical insights.
