## IMA Postdoc Seminar (September 23, 2008)

**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.

**Slides:** PDF