Department of Mathematics

Speaker: Mark Ablowitz
Title: Nonlinear Waves from Beaches to Photonic Lattices
Affiliation: University of Colorado, Boulder
Date: Friday, February 8, 2013
Place and Time: Room 101, Love Building, 3:35-4:30 pm
Refreshments: Room 204, Love Building, 3:00 pm

Abstract. The study of localized waves has a long history dating back to the discoveries in the 1800s describing solitary water waves in shallow water. In the 1960s researchers found that certain equations, including the 1+1 dimensional Boussinesq and Korteweg-deVries (KdV) equations frequently arise. These equations admit localized solitary wave-or soliton solutions. They have 2+1 dimensional counterparts: Benney-Luke (BL) and Kadomstsev-Petviashvili (KP). Some of the two-dimnsional solutions will be discussed as well as ocean observations and videos. In optics and Bose-Einstein condensation nonlinear Schrodinger type equations arise frequently. In photonic lattices with simple periodic strong potentials, discrete and continuous NLS equations arise. In non-simple periodic, hexagonal or honeycomb lattices, discrete Dirac-like systems and their continuous analogs can be derived. They are found to have interesting properties. Since honeycomb lattices occur in the material graphene, the optical case is often termed photonic graphene.