MATHEMATICAL PHYSICS WEBINAR
RUTGERS UNIVERSITY
__________________________________________
Panayotis Kevrekidis – University of Massachusetts, Amherst
Wednesday, November 6th , 10:45AM EDT
Teaching an old dog some new tricks: from discrete solitons and vortices,
to rogue waves, flat bands and PINNs in variants of the discrete nonlinear Schrodinger model
In this talk we will revisit nonlinear dynamical lattices, starting with an overview of their physical applications in atomic, optical, mechanical and metamaterial systems. Then we will focus most notably on the prototypical discrete nonlinear Schrodinger (DNLS) model. We will explore at first some of its main features, including discrete solitons, discrete vortices and related structures in 1d, 2d and 3d, not only in square, but also in other lattice patterns (hexagonal, honeycomb, etc.).
Then, time permitting, and motivated by recent experimental and mathematical developments, we will consider some interesting recent variants on the theme, such as, e.g., what happens in Kagome' lattices. This will motivate a discussion of the notion of so-called flat bands and compactly supported nonlinear states therein. We will discuss simple methods of producing lattices with flat bands, and some experimental implementations thereof in electrical circuits. Compactly supported nonlinear states will also be seen to arise from nonlinearly dispersive variants of the model recently proposed in the context of the mathematical analysis of turbulent cascades. We will also touch upon extreme events and so-called rogue waves in integrable and non-integrable variants of the model. We will end with some machine-learning inspired touches of how to "discover" such lattices from data, using Physics-Informed Neural Networks (PINNs) and how to improve PINNs using the symmetries of the lattice.