This project will be a study of the geometric and combinatorial (i.e. enumerative) structures of soliton solutions of certain integrable equations. Those integrable equations appear as models of a number of important physical systems. Current methods for studying soliton solutions of these systems focus on analytic and algebraic analysis of the solutions. Typical goals are finding exact formulas for solutions and examining the stability of solutions. Recently it has been seen that in the connections between these integrable systems and theoretical physics, the geometry and combinatorics of the systems play a crucial role in understanding their solutions. We will categorize and enumerate the number and type of soliton solutions of the equations of Kadomtsev-Petviashvili (KP)-type. The KP equation is a two-dimensional nonlinear wave equation which can be used to describe shallow water surface waves (e.g. beach waves). The KP equation posses various types of soliton solutions, and they show complicated interaction patterns due to the nonlinearity of the equation. Understanding of this complexity in the solution pattern is the main goal of the project. More generally equations of KP-type have been connected to the families of random matrix models of theoretical physics, and the results of the project provide geometric and combinatorial structures of the models.
The result of the project will lead to a deeper understanding of the soliton solutions of the KP-type equations. In the process, we will consider a number of new and interesting combinatoric problems. The KP-type equations are important integrable systems which capture much of the general structure of such systems; the results and the techniques developed to find them will influence the combinatoric study of multiple soliton solutions of other integrable systems. The results, from the point of view of random matrices, will be of interest in theoretical physics, combinatorics, and integrable systems. The project includes educational aspects through collaboration between senior and junior faculty and graduate students in several different areas of mathematics and physics, including applied math, algebraic geometry, combinatorics, fluid mechanics and high-energy physics.
