Department of Applied Mathematics
University of Colorado, Box 526
Boulder, CO 80309-0526
 Telephone: 303-492-5502, Fax: 303-492-4066
Email: mark.ablowitz@colorado.edu
 

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.