Speaker: Xianjin Chen (IMA)
Title: Two Stable Methods for Multiple Unstable Solutions to Semilinear Variational Elliptic Systems
Abstract: Exhibiting many novel new phenomena that are not present in the single equation case, systems are much more interesting in many applications. Motivated by the growing experimental observations and studies of various nonlinear vector phenomena (e.g., spatial vector solitons) arising in diverse physical contexts (e.g., condensed matter physics, nonlinear optics, etc), the speaker will give an overview of some computational theory and methods for finding multiple unstable solutions (e.g., saddle points) to three types of nonlinear variational elliptic systems: cooperative, noncooperative, and Hamiltonian. In particular, two local characterizations of multiple unstable solutions to variational elliptic systems as well as two stable methods (called the local min-orthogonal method and the local min-max-orthogonal method) for finding saddle points of finite or infinite Morse index will be presented. Finally, both methods were applied to solve elliptic systems of those three types mentioned for multiple unstable solutions.