Combinatorial and geometrical aspects of integrable systems and their applications to physics

这个项目将研究某些可积方程的孤子解的几何和组合（即枚举）结构。这些可积方程是许多重要物理系统的模型。目前研究这些系统孤子解的方法主要集中在解的解析和代数分析上。典型的目标是找到解的精确公式并检查解的稳定性。最近人们发现，在这些可积系统与理论物理之间的联系中，系统的几何和组合学在理解它们的解方面起着至关重要的作用。我们将对Kadomtsev-Petviashvili （KP）型方程的孤子解的数目和类型进行分类和列举。KP方程是一个二维非线性波动方程，可用于描述浅水表面波（如滩波）。KP方程具有多种类型的孤子解，由于方程的非线性，它们表现出复杂的相互作用模式。理解解决方案模式中的这种复杂性是该项目的主要目标。更一般的kp型方程已经与理论物理的随机矩阵模型族联系起来，项目的结果提供了模型的几何和组合结构。该项目的结果将导致对kp型方程的孤子解的更深层次的理解。在这个过程中，我们将考虑一些新的和有趣的组合问题。kp型方程是重要的可积系统，它捕获了此类系统的大部分一般结构；这些结果和为找到它们而开发的技术将影响其他可积系统的多孤子解的组合研究。结果，从随机矩阵的角度来看，将对理论物理，组合学和可积系统感兴趣。该项目包括教育方面，通过在数学和物理的几个不同领域，包括应用数学、代数几何、组合学、流体力学和高能物理的高级和初级教师和研究生之间的合作。

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.