Mathematical Sciences: Waves and Stability in Nonlinear Partial Differential Equations

9403871 PEGO非线性波在科学和工程中重要的偏微分方程组中扮演着重要的角色。这项拟议工作的一个主要目标是发展有效的方法来了解许多这样的波的稳定性特性。这位提出者和合作者最近介绍了一些很有前途的新方法来研究孤立波简化模型的稳定性。我们计划将这些方法扩展到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。第二个主要研究课题将是研究液气临界点附近的流体中的低速流动，无论是在零重力条件下还是在1-G条件下。高可压缩性和反常的物理性质是这种状态的特点，在航天器和地球上的实验中观察到了一些现象，这些现象无法解释或只能部分解释。所提出的稳定性分析涉及到几个数学工具的使用和进一步发展，包括埃文斯函数和Krein摄动行列式。两者都是解析函数，其零点对应于本征值或共振极点。对于色散方程的孤立波，将进一步发展线性稳定性分析来证明非线性稳定性。此外，还将研究一种重新表述哈密顿系统的新方法，以使约束成为守恒量：水波、缆索在零重力中的弯曲以及在一个方向上具有刚性的各向异性材料。关于燃烧中“零马赫数流动”的应用数学的最新发展将被适应并推广到研究近临界纯流体。

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.