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

本计画将研究某些可积方程式之孤立子解之几何与组合（即计数）结构。 这些可积方程作为许多重要物理系统的模型出现。 目前研究这类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决方案的精确公式和检查解决方案的稳定性。 最近人们发现，在这些可积系统和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。 本文将对Kadomtsev-Petviashvili（KP）型方程的孤子解的个数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（例如海滩波）。 KP方程包含了各种类型的孤子解，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。更一般地说，KP型方程已经与理论物理中的随机矩阵模型族联系起来，本项目的结果提供了模型的几何和组合结构，将有助于对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，它捕捉了此类系统的一般结构，其结果和发现方法将影响其他可积系统多孤子解的组合研究。 结果，从随机矩阵的角度来看，将感兴趣的理论物理，组合学和可积系统。 该项目包括教育方面，通过高级和初级教师和研究生之间的合作，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.