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Some Case Studies of Nonlinear Dynamical Lattices: From the Discrete Nonlinear Schrodinger Equation to PT-Symmetric Oligomers

Tuesday, December 4, 2012 - 9:00am - 9:50am
Keller 3-180
Panayotis Kevrekidis (University of Massachusetts)
In this talk, we 'll start by reviewing some of the developments on nonlinear
dynamical lattices of the discrete nonlinear Schrodinger type. We will explore
ideas of continuation from the so-called anti-continuum limit, in order to
identify discrete solitons and their stability in 1d lattices, as well as
discrete vortices and more complex entities (such as vortex cubes) in two-dimensional and three dimensional case examples. Time-permitting we will
present some extensions of these nearest-neighbor lattices to longer range
interaction examples and how similar ideas carry over to the setting of
Klein-Gordon lattices. More importantly, we 'll venture to go beyond the
Hamiltonian realm to a setting which has recently gained significant momentum
in the physical community but has very slightly been touched upon in the
mathematical literature, namely that of the PT-symmetric lattices. The latter
are, in a sense, a very special case example that stands between the Hamiltonian
and the dissipative case. We will attempt to illustrate via some prototypical case
examples how what we know from the Hamiltonian case is drastically modified
in this PT-symmetric setting, highlighting some of the emerging mathematical challenges in this field.
MSC Code: 
70Kxx