|Mark J. Ablowitz
Professor of Applied Mathematics
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. 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 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.
This site was last updated June, 2007