Nonlinear evolution equations, asymptotics and applications

非线性色散波最常见的表现形式可能是破碎的海浪。 然而，非线性色散波在自然界中无处不在，出现在从水波到光学、声学、铁磁学、凝聚态、宇宙学等许多领域。 当耗散和非线性是系统中的主要物理效应时，小耗散的状态通常会引起冲击波（最常见的表现是音爆）。 当用弥散代替耗散时，类似的现象就是弥散激波的现象. 这些都是非平稳相干，多尺度振荡结构。 引起色散冲击波的物理介质从水波到超流体、非线性光子学和磁自旋系统。 例如，在流体动力学中，分散冲击被称为波状孔。 尽管在过去的50年里，人们做了大量的工作来理解这些现象，但许多基本问题仍然存在。 该项目的第一个组成部分是开发数学工具，用于研究某些非线性演化方程的解的行为，这些方程描述了一个以上空间维度的非线性波动现象。 该项目的第二部分涉及应用这些工具来研究各种感兴趣的领域，从水波到光学，网络和统计物理。 最后，该项目还将作为培养几个博士的工具。对引起频散冲击的非线性介质的数学研究常常导致某些双曲守恒律系统。 在过去的五十年里，各种方法已经成功地应用于研究这类系统。 然而，色散非线性波动方程在多维空间中的解的行为并没有得到很好的理解。 特别是，在两个空间维度中的分散冲击的形成和传播的数学表征在很大程度上仍然是一个悬而未决的问题。 PI最近的工作为研究这方面的一些长期未决问题开辟了新的途径。 具体而言，该项目包括四类问题：（一）使用Kadomtsev-Petviashvili方程的Whitham调制方程（所谓的KP-Whitham方程，最近由PI推导出）来研究分段常数孤子初始数据的时间演化以及2+1维色散激波的形成和动力学。 (ii)研究上述KP-Whitham方程的可积结构及其精确解。 (iii)各种（2+1）维非线性薛定谔型发展方程的Whitham调制方程的推导及应用。 (iv)应用小色散极限和Whitham调制理论描述p-星形网络和某些随机矩阵模型的相态图。 该项目的首要主题是提高我们对多个空间维度的色散波现象的理解。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The most familiar manifestation of nonlinear dispersive waves is perhaps that of a breaking ocean wave. However, nonlinear dispersive waves are ubiquitous in nature, appearing in many fields ranging from water waves to optics, acoustics, ferromagnetics, condensed matter, cosmology and beyond. When dissipation and nonlinearity are the dominant physical effects in a system, the regime of small dissipation often gives rise to shock waves (the most familiar manifestation being a sonic boom). The analogous phenomenon when dissipation is replaced by dispersion is that of dispersive shock waves. These are non-stationary coherent, multiscale oscillatory structures. Physical media giving rise to dispersive shock waves range from water waves to superfluids, nonlinear photonics, and magnetic spin systems. For example, in fluid dynamics, dispersive shocks are known as undular bores. Even though much work has been done over the past fifty years to understand these phenomena, many fundamental questions remain. A first component of this project involves the development of mathematical tools for studying the behavior of solutions of certain nonlinear evolution equations describing nonlinear wave phenomena in more than one spatial dimension. The second component of the project involves the application of these tools to study a variety of areas of interest, ranging from water waves to optics, networks, and statistical physics. Finally, the project will also serve as a vehicle for training several Ph.D. students.The mathematical study of nonlinear media giving rise to dispersive shocks often leads to certain systems of hyperbolic conservation laws. Over the last fifty years, various methods have been applied with success to study these kinds of systems. However, the behavior of solutions of dispersive nonlinear wave equations in more than one spatial dimension is not as well understood. In particular, a mathematical characterization of formation and propagation of dispersive shocks in two spatial dimensions is still largely an open problem. Recent work by the PI has opened up new avenues to study some long-standing open problems in this regard. Specifically, this project comprises four classes of problems: (i) Use of the Whitham modulation equations for the Kadomtsev-Petviashvili equation (the so-called KP-Whitham equations, which were recently derived by the PI) to study the temporal evolution of piecewise-constant soliton initial data and the formation and dynamics of dispersive shock waves in 2+1 dimensions. (ii) Study of the integrability structure of the above-mentioned KP-Whitham equations and their exact solutions. (iii) Derivation and application of Whitham modulation equations for various (2+1)-dimensional evolution equations of nonlinear Schrodinger type. (iv) Application of small dispersion limits and Whitham modulation theory to characterize phase state diagrams of p-star networks and certain random matrix models. The overarching theme of the project is to advance our understanding of dispersive wave phenomena in more than one spatial dimension.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.