Geometric Combinatorics and Hypergeometric Functions in Integrable Systems and Their Physical Applications

这项研究项目是为了研究某些类型的非线性微分方程类，它是对几种物理现象的数学描述。该项目包括对非线性波的研究，特别是在浅水中。由于非线性，当一个波叠加在另一个波上时，波不仅仅是“相加”的，这在线性情况下是真的，而是相互作用的。这种波形成复杂的波型，在产生大幅度波方面有非常重要的物理应用，例如在海滩附近的浅水或海啸中。了解这种极端海浪的性质和动态，特别是在人口稠密地区附近，是一项重大而紧迫的任务。该项目还旨在研究经典特殊函数和一类描述例如气体动力学中的冲击波现象的流体动力学系统之间的新联系。研究活动将涉及研究生，他们将接受纯数学和应用数学领域的培训，并将获得第一手研究经验。此外，这些工作的结果将有助于研究其他物理问题中出现的类似的非线性波动现象，包括非线性光学、粒子物理和等离子体。PI将继续他在某些二维可积系统的几何和组合方面的工作，特别是Kadomtsev-Petviashvili(KP)方程。KP方程允许一大类具有复杂二维波型的孤立波解，这些孤立波解有时被称为KP网孤子。这个项目还旨在进一步研究PI最近关于一类与定义在Grassman上的广义超几何函数有关的可积流体动力学系统的研究。这个项目有两个主要目标，即(1)利用几何和组合理论研究Kp网络-孤子的复杂二维图样的详细结构；(2)构造和分类由广义超几何函数生成的可积流体动力学系统。此外，该项目还包括正在与实验者合作，以便将所获得的关于KP孤子的理论结果与在水波水箱中进行的实验室实验进行比较。

This research project is to investigate certain classes of nonlinear differential equations arising as a mathematical description of several physical phenomena. The project includes an investigation of nonlinear waves, particularly, in shallow water. Because of nonlinearity, when one wave is superimposed on the other the waves are not just "additive," as would be true in the linear case, but interact with each other. Such waves form complex wave patterns and have very important physical applications in the generation of large-amplitude waves, for example, in shallow water near a beach, or tsunamis. Understanding the nature and dynamics of such extreme waves, especially near highly populated areas is a significant and urgent task. The project also aims to investigate a new connection between classical special functions and a class of hydrodynamic systems which describe, for example, shock wave phenomena in gas dynamics. The research activities will involve graduate students who will be trained in the fields of both pure and applied mathematics, and will gain first-hand research experience. It is further anticipated that the results from the proposed work would be useful in the study of similar nonlinear wave phenomena that occur in other physical problems including nonlinear optics, particle physics and plasmas.The PI will continue his work on geometric and combinatorial aspects of certain two-dimensional integrable systems, particularly, the Kadomtsev-Petviashvili (KP) equation. The KP equation admits a large class of solitary wave solutions with complex two-dimensional wave patterns, which are sometimes referred to as the KP web-solitons. This project also aims to give a further investigation of the PI's recent study on a class of integrable hydrodynamic systems in connection with the generalized hypergeometric functions that are defined on the Grassmannian. This project has two main goals, namely, (1) to investigate the detailed structure of the complex two-dimensional patterns of the KP web-solitons using geometric and combinatorial theories, and (2) to construct and classify integrable hydrodynamic systems generated by the generalized hypergeometric functions. Also, the project includes ongoing collaboration with experimentalists in order to compare the obtained theoretical results on the KP solitons with laboratory experiments performed in water wave tanks.