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(即可计数)结构。这些可积方程似乎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分析和代数分析上。典型的目标是找到精确的解的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)型方程的孤子解的个数和类型进行分类和计数。KP方程是一种二维非线性波动方程，可以用来描述浅水表面波(如海滩波)。KP方程具有多种类型的孤子解，由于方程的非线性，孤子解呈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。更广泛地说，KP型方程被连接到理论物理的随机矩阵模型族中，该项目的结果提供了模型的几何结构和组合结构，该项目的结果将导致对KP型方程孤子解的更深入的理解。在这个过程中，我们将考虑一些新的和有趣的组合问题。KP型方程是一类重要的可积系统，它反映了这类系统的一般结构，其结果和寻找它们的技巧将影响其他可积系统的多孤子解的组合研究。从随机矩阵的观点来看，这些结果将在理论物理、组合学和可积系统中引起人们的兴趣。该项目包括高级和初级教师以及研究生在数学和物理的几个不同领域的合作，包括应用数学、代数几何、组合数学、流体力学和高能物理。

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.