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. In the 1970s we showed how IST can be applied to a large class of nonlinear PDEs
(often called AKNS systems) and nonlinear discrete systems (Ablowitz Ladik). IN 1980s we extended this
to integro-differential equations and 2+1 systems such as Kadomtsev-Petviashvili and Davey Sewartson equations. We found how lump type solutions to the KPI equation fit into the IST framework and in late 1990s early 2000s extended these lump solutions to a much wider class of solutions.
In 1990s-2000s we showed that the four dimensional self-dual Yang-Mills (SDYM) equations can be reduced to certain novel ODE's including the Darboux-Halphen-Chazy equations. These systems also relate to ones studied by Ramanujan.We also showed how classical 1+1, 2+1 dimensional systems can be found by reduction from SDYM. In the 2000s we showed how 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. In 2010s we showed that line soliton type strcutures occur every day on nearly flat beaches. In 2013--2017 we found a large new class of integrable nonlocal wave equations. These new equation have remarkably simple structure. In 2019 we showed that integrable nonlocal nonlinear Schrodinger systems can be found by asymptotic reduction from KdV, nonlinear Klien-Gordon and water waves. In fact this reduction is universal. In 2018 we found solutions to nonlocal problems which decay and whose amplitudes are constant at infinity.
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.
In 2010s we showed that line soliton type strcutures occur every day on nearly flat beaches.