Photonic Lattices/ Photonic Graphene: The nonlinear Schrodinger equations with a two dimensional lattice background potential has attracted wide interest. A problem of keen interest is to understand the wave propagation in a direction orthogonal to the background lattice. When the background is a hexagonal or honeycomb (HC) potential it is often referred to as photonic, or optical graphene. This name is due to the fact that the material graphene also has such hexagonal background structure. In our papers we have shown, in the tight binding limit, how to transform the study of the wave phenomena to a coupled nonlinear discrete system. In unbounded HC lattices the associated Brillouin surfaces may touch at certain points. These points have conical type singularities and the wave structure leads to ``conical diffraction’’. Under strain, in the bulk there can be elliptical and straight line diffraction. In our papers we have found reduced PDEs which describe these situations. If the HC lattice has an edge then localized linear and nonlinear states, stationary or traveling, can be found. We have shown that the envelope of the linear/nonlinear edge states satisfy one plus one dimensional classical linear/nonlinear Schrodinger (NLS) equations. In the focusing regime the NLS equation has soliton solutions. These solitons can be transmitted or reflected across the boundaries and can inherit the role of linear topological insulators.
Dynamics of ultra-short pulses in mode-locked lasers:
Mathematical models of mode-locked lasers, such as Ti:sapphire lasers operating
in the femto-second regime are being studied. These models admit localized
solitary wave solutions, usually termed solitons by researchers in this
field. Improved models describing the dynamics of such laser pulses are being
developed. Some of the key concepts such as dispersion management also arise in
the propagation of nonlinear pulses in optical fibers. This is an area in which
our group has considerable expertise.
Ultra-high bit rate communication in optical fibers:
This includes dispersion managed transmission processes, soliton and
quasi-linear pulse transmission, frequency and timing shift analysis,
differential phase shift key transmission, addition of suitable filters and
control mechanisms and dynamics originating from randomness of lengths or
Nonlinear optical waveguides: Dynamics and propagation
of pulses in regular and irregular lattice systems; discrete optical soliton
transmission; analysis of the effects transverse wave guide dynamics;
comparisons of discrete and continuous models; exact traveling waves in
discrete models; solitons in diffraction managed waveguides.
Dispersive Shock Waves (DSW's): Classical shock waves
arise in systems where the viscosity is relatively small. Another type of shock
wave can exist in systems where the dispersion, rather than the viscosity, is
small. The propagation of dispersive shock waves follows a different scenario
from classical shock waves. The interaction of DSW's are being carefully
investigated. Applications include the hydrodynamics of Bose-Einstein
condensates. Recent experimental studies by researchers have demonstrated these
novel fluid dynamical properties.
Water waves: Motivated by theoretical work done by our
group, experimentalists at Penn State University have shown that periodic water
waves in deep water can exhibit chaotic dynamics. While envelope solitons are
experimentally reproducible, this is not always the case for envelopes
associated with the classical periodic water waves. Recently, we have also been
working on a novel formulation of water waves. A related issue being
investigated includes the study of the propagation of lumps in multi-dimensional
Solutions of nonlinear equations by IST: The Inverse Scattering Transform (IST) is being employed to study certain nonlinear equations in 2+1 and 1+1 dimensions. New solutions and boundary value problems are being analyzed. We have shown that the four dimensional self-dual Yang-Mills equations can be reduced to certain novel ODE's including the Darboux-Halphen-Chazy equations. These systems also relate to ones studied by Ramanujan. Certain continuous and discrete vector nonlinear Schrodinger systems can be analyzed by the IST method. The method allows one, amongst other things, to understand the character of vector soliton collisions. In this research we also analyze the continuous and discrete matrix direct and inverse scattering problems. We also analyze boundary value problems which have non-vanishing data at infinity.