Applied Mathematics Colloquium
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, such as the Korteweg-deVries (KdV), Kadomstsev-Petviashvili and nonlinear Schrodinger (NLS) equations arise widely. Both equations admit localized solitary wave--or soliton solutions. Employing a nonlocal formulation of water waves interesting asymptotic reductions of water waves are obtained. Some solutions will be discussed as well as how they relate to ocean observations. In the study of photonic lattices with simple periodic potentials, discrete and continuous NLS equations arise. In non-simple periodic, hexagonal or honeycomb lattices, novel discrete Dirac-like systems and their continuous analogs can be derived. They are found to have interesting properties. Honeycomb lattices also occur in the material graphene; the optical case is termed photonic graphene.