The research supported by this award will address aspects of nonlinear waves and soliton theory which have not be fully understood before. In particular, it is to develop new analytical paradigms for solving nonlinear problems. This research will develop an Inverse Scattering Transform (IST) for general initial value problems of the Camassa-Holm and related equations. Current techniques only allow one to treat a limited class of initial value problems. For the Camassa-Holm Equation, which is an important limit of a class of hydrodynamical problems, only positive potentials can be treated. However, the prediction of the evolution involving negative potentials are also quite important. Such potentials are known to model wave-breaking. This research will expand the power of the IST method to those cases where the potential could be negative and/or positive in any region. The techniques developed by this research are important because they then could be applied to other important and related, but more complex systems. One related and important nonlinear optical system is called "Degenerate Two-Photon Propagation", which is a very fast nonlinear interaction, by which one can generate intense second harmonic radiation from a laser beam. The latter system has potential applications to future nonlinear optical systems, including optical logic circuits and switches. This award will also support nonlinear wave research for finding methods and means for modeling nonintegrable physical systems which do contain solitary waves.
This award is to support research to extend our mathematical methods into physical regimes which have remained unsolvable. It will center on a solution method for an equation which describes how the surface of shallow water behaves when disturbed. The reason for the choice of this equation is not necessarily just for increasing our understanding of surface water waves, although that will occur. Rather, the choice is made because this equation is relatively simple in its structure, and furthermore, is at the central core of a class of important physical systems. Developing methods for solving this core system will allow one to immediately apply the same techniques to these other systems. One of these systems, mentioned above, is from optical physics. Another such system is also found in optical physics, but also occurs in other areas of physics as well. It is an interaction whereby a laser beam, propagating in a medium or a plasma, can go unstable, or could stabilize. As a product of this research, we would then have methods for predicting when and how the above mentioned interaction would produce a stability or an instability. The results from these proposed research topics will have a broader impact than simply the mathematics that are proposed. They will expand the physical systems that can be studied mathematically, will provide new areas for mathematical research, and will further the understanding of the physics behind the physical systems so studied.
