Mathematical Sciences: Waves and Stability in Nonlinear Partial Differential Equations

[403871]非线性波在科学和工程上重要的偏微分方程组中起着重要的作用。提出的工作的一个主要目标是开发有效的方法来理解许多这样的波的稳定性。提议者和合作者最近介绍了一些有前途的新方法来研究孤立水波的简化模型的稳定性。我们计划扩展这些方法来研究各种重要系统中的波，包括完整的水波方程和粒子物理和相变的简化模型。研究的第二个主要课题将是研究在零重力和一重力条件下液气临界点附近流体的低速流动。高压缩性和异常的物理性质是这种状态的特征，在宇宙飞船和地球上的实验中观察到一些无法解释或只能部分解释的现象。所提出的稳定性分析涉及使用和进一步发展几个数学工具，包括埃文斯函数和克莱恩的扰动行列式。两者都是解析函数，其零点对应于特征值或共振极点。利用线性稳定性分析来证明色散方程孤立波的非线性稳定性将得到进一步发展。此外，一种重新表述哈密顿系统的新方法，使约束成为守恒量，将被研究：水波，零重力下电缆的弯曲，以及在一个方向上“僵硬”的各向异性材料。应用数学中关于燃烧中“零马赫数流动”的最新发展将适用于研究近临界纯流体。

9403871 Pego Nonlinear waves play a significant role in systems of partial differential equations of importance in science and engineering. A primary goal of the proposed work is to develop effective methods for understanding the stability properties of many such waves. The proposer and collaborators have recently introduced some promising new methods for studying the stability of simplified models of solitary water waves. We plan to extend these methods to study waves in various systems of importance, including the full water wave equations and simplified models of particle physics and phase transitions. A second major topic of investigation will be the study of low velocity flow in fluids near the liquid-gas critical point, both in zero-G and 1-G conditions. High compressibility and anomalous physical properties characterize this regime, and several phenomena have been observed in experiments in spacecraft and on earth which are unexplained or only partly explained. The proposed stability analyses involve the use and further development of several mathematical tools, including the Evans function, and Krein's perturbation determinant. Both are analytic functions whose zeros correspond to eigenvalues or resonance poles. The use of linear stability analysis to prove nonlinear stability will be further developed for solitary waves of dispersive equations. Also, a new way of reformulating Hamiltonian systems, so that constraints become conserved quantities, will be investigated for: water waves, flexure of cables in zero-G, and anisotropic materials that are `stiff' in one direction. Recent developments in applied mathematics regarding `zero-Mach number flows' in combustion will be adapted and extended for studying near-critical pure fluids.