Novel Challenges in Nonlinear Waves and Integrable Systems

This research concerns nonlinear wave phenomena, with particular emphasis on the study of a class of nonlinear evolution equations commonly referred to as soliton equations or integrable systems. These systems model physical phenomena in diverse fields such as deep-water waves, plasma physics, nonlinear optical fibers, low temperature physics and Bose-Einstein condensates, and magneto-static spin waves. Some of the equations studied belong to the class of the nonlinear Schrödinger (NLS) equation and its various generalizations, which constitute universal models for weakly dispersive nonlinear wave trains. Others belong to the complex short-pulse equation and its generalizations, which describe the propagation in nonlinear media of ultra-short optical pulses. These have applications to laser material processing, laser microscopy, optical clocks and measurements, medicine (e.g., eye surgery and vision correction), and telecommunication. A distinguishing feature of these systems is that, in addition to standard solitons, localized traveling waves that preserve their shape and velocity, they also admit so-called loop solitons, which are not single-valued solitons, as well as solutions that oscillate between the two. Another phenomenon that this research aims at elucidating is the formation of rogue waves. These extreme events are unusually large surface waves with wave crests up to four times the mean level that appear from nowhere and disappear without a trace and can therefore cause significant damage to ships, and offshore drilling units. This research addresses the role of solitons and modulational instability as possible concurrent mechanisms for the formation of rogue waves. Rogue waves have also been recently observed experimentally in optical fibers. The training of graduate students at the University at Buffalo, and outreach activities aiming at promoting the growth of women in applied mathematics will be integral components of this project. At a more technical level, the investigation deals with certain integrable systems in situations in which the boundary conditions play a key role, as well as various kinds of singular limits. Specific tasks include: (i) the development of the inverse scattering transform for various NLS systems with non-zero boundary conditions at infinity, and in particular with boundary conditions corresponding to counter-propagating waves; (ii) the investigation of concrete questions from applications in Bose-Einstein condensates (e.g., trains of solitons, bound states, windings) and in nonlinear optics (nematic liquid crystals, etc); (iii) the rigorous study of the discrete spectrum and spectral singularities of NLS systems; (iv) development of the inverse scattering transform for the complex coupled short-pulse equation; (v) the investigation of solitons, soliton interactions and rogue waves for the above systems, and applications to non-integrable systems in regimes close to the integrable ones; and (vi) the study of phase transitions in networks as critical behavior of solutions of soliton equations. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.