Collaborative Research: Nonlinear Dynamics and Spectral Analysis in Dispersive Partial Differential Equations

深水内波、等离子体中的离子声波、激光在高折射率材料中的传播以及在极低温度气体中的集体粒子行为都是物理系统，其时间行为都可以用非线性色散波动方程来描述。术语“非线性”指的是大小相关的响应特性，术语“色散”指的是波的波动如何影响运动的速度和方向。在数学层面上，人们研究这些方程的通解和特殊类型解的行为，并试图提供在物理环境中观察到的现象的定量描述。在这个项目中，研究人员探索了相干波是如何传播的--无论它们是在遇到障碍时保持形状，还是分裂和溶解，还是坍塌成奇点。每一种可能性都取决于方程的非线性和色散特性；在过去的几十年里，已经发展了数学技术来模拟和测量这些特性。该项目旨在改进现有方法，并将这些方法应用于新的方向。该项目包含了不同数学学习水平的教育努力。这些措施包括为本科生和研究生以及博士后学者提供建议，特别强调吸引代表性不足的少数群体。这项研究的主要目的将是为某些类型的非线性色散方程的解的行为提供分析性描述。所考虑的两类主要方程是KdV(KdV)族和NLS(NLS)族。这两类方程都满足被称为局部维里估计的强大色散估计，此外，KdV族还满足控制质量运动的单调性。多尺度方法、谱分析、色散估计的应用和单调界是将被利用和推广的核心方法。确定了几个焦点问题，其中一些经典特征，如标度不变性或局部影响的性质，已被移除或削弱，这为发展新技术提供了刺激，同时也为真实物理现象提供了新的描述。主要感兴趣的现象是当孤立波和线孤子等相干结构相互作用及其环境时的动力学，以及对奇异坍塌解如何产生及其渐近描述的描述。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Internal deep-water waves, ion-acoustic waves in a plasma, laser propagation in highly refractive materials, and collective particle behavior in very low temperature gases are all physical systems whose behavior in time is modeled by a nonlinear dispersive wave equation. The term "nonlinear" refers to a property of size-dependent response and the term "dispersive" refers to how the fluctuations of the wave influence the speed and direction of motion. At the mathematical level, one studies the behavior of general solutions and special types of solutions to these equations and seeks to provide quantitative descriptions of phenomena observed in the physical setting. In this project, the investigators explore how coherent waves travel - whether they retain their shape despite encountering obstacles, break apart and dissolve, or collapse into a singularity. Each of these possibilities hinges on both the nonlinear and the dispersive character of the equations; over the past several decades, mathematical techniques have been developed to model and measure these properties. The project aims to improve existing methods and apply the methods in new directions. The project contains educational efforts at various levels of mathematical learning. These include advising undergraduate and graduate students as well as postdoctoral scholars, with special emphasis on attracting underrepresented minorities. The main objective of the research will be to provide analytical descriptions of the behavior of solutions to certain classes of nonlinear dispersive equations. The two main categories of equations considered are the Korteweg-de Vries (KdV) family and the nonlinear Schrödinger (NLS) family. Both classes of equations satisfy powerful dispersive estimates called local virial estimates, and the KdV family in addition satisfies a monotonicity property that controls the movement of mass. The multiple-scale method, spectral analysis, application of dispersive estimates, and monotonicity bounds are core methods that will be utilized and extended. Several focus problems are identified in which some classical feature, like scale-invariance or a property of localized influence, have been removed or weakened, providing the stimulus to develop new techniques, while at the same time, provide new descriptions of real physical phenomena. The phenomena of primary interest are the dynamics of coherent structures like solitary waves and line solitons as they interact with each other and their environment, and the description of how singular collapsing solutions arise and their asymptotic description.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.