Research Interests

Photonic Lattices/ Photonic Graphene/Topological Insulators: 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. These unidirectional waves have been shown to possess topological properties; the material is called a topological insulator: one which acts as an insulator in its interior and has conducting states on its edge or surface. The edge waves propagate without reflection and propagate stably around defects. Using tight binding analysis and asymptotic analysis 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 this NLS equation can possess soliton solutions. These solitons can be transmitted or reflected across the boundaries and can inherit the role of linear topological protected waves. In recent work we have found lattices and tight binding equations and associated topological modes for a wide class of lattice systems where each sublattice is allowed to vary along its own trajectory.

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 dispersion.

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. Experimental studies by researchers have demonstrated these novel fluid dynamical properties. Recent work has found modulation equations for 2+1 dimensional systems. These new equations open the way to studies of 2+1 DSW systems.

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 water waves.

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.  Recently we have found new classes of nonlocal equations which are integrable. Many of these new equation have remarkably simple structure. Via IST we have found solutions to problems which decay and whose amplitudes are constant at infinity.