9703850 Ablowitz Research involving a class of physically significant nonlinear systems arising in the study of nonlinear wave motion is proposed. Included in the studies are a new class of localized multidimensional solutions which can be related to a topological number; i.e. an index or winding number. Associated with the self dual Yang-Mills equations is a novel class of nonlinear ordinary differential equations, referred to as Darboux-Halphen (DH) type systems. These equations can be integrated by associating them with, and analyzing, certain linear monodromy problems. The linear problems have monodromy data which are evolving. The solutions of DH systems can be extremely complicated, exhibiting natural boundaries and dense branching in the complex plane. In special cases the solutions are expressed in terms of automorphic functions. Frequently researchers use numerical simulation to obtain approximations to solutions of nonlinear wave equations. Research by the PI and collaborators has demonstrated that computational chaos can be induced from truncation and roundoff errors. Use of integrable systems is extremely important since analytical results can be compared with numerical simulation. The study of nonlinear wave equations has wide ranging applications in physics and engineering. Intimately related to the studies in this proposal is a class of equations, called integrable systems. The mathematics of integrable systems and their solutions have been shown to play an important and fundamental role in our understanding of a variety of real world phenomena. Integrable systems usually admit a special class of extremely stable, localized wave solutions, referred to as solitons, which arise frequently in physical contexts. For example, in the study of fluid dynamics, huge soliton internal ocean waves, sometimes a mile in length, and hundreds of feet in amplitude, have been located in many parts of the ocean; e.g. near the coast of Oregon. Such waves can be devastating to anything caught in their path, hence knowledge of where such internal solitons are located in the ocean is of interest. Another application of solitons is in the study and designing of efficient long distance communication systems. Solitons eliminate the need for electronic signal regenerators which can seriously inhibit data rates. Researchers have used solitons successfully in the laboratory and currently they are being used in prototype communication systems. It may well be that future high data rate communications systems will be designed to propagate multiple soliton waves.
