Mathematical Sciences: Nonlinear Hyperbolic and Dispersive Waves

9311438 Rosales对色散和双曲型方程组的非线性波的研究。主要研究项目有：(1)“弱激波的聚焦与反射与冯·诺依曼悖论”：研究弱非线性几何光学理论中出现的这些问题及相关问题；(2)“双曲系统中的共振周期波”；(3)“闭管中的共振振荡”。利用WNGO，将获得和研究气体动力学方程和其他双曲系统的一些有趣和令人惊讶的解(4)“铁磁流体中的弱非线性长表面波”：铁磁流体中表面波的简化方程（使用长波渐近性）将得到并研究(5)“非线性色散系统中的波调制击穿”(6)“方波爆轰”：采用简化模型，研究方波爆轰理论中产生“灾难性不稳定性”的原因和机理。本提案涉及非线性波的研究。波浪在声学、光学、气体动力学、燃烧理论、海洋动力学、固体力学等许多领域都有重要的应用。在过去的几十年里，非线性波理论取得了许多进展，但仍然存在许多难题。这一提议中波动理论问题的一个例子是弱激波反射和聚焦的“冯·诺伊曼悖论”。自从约翰·冯·诺伊曼（John von Neumann）指出这个问题以来，它一直是许多研究的对象，但仍然没有定论。私家侦探和一名研究生在先前的研究中取得了重大进展，并且非常接近于完全解决问题。这个悖论的解决将具有相当大的科学意义，并将阐明许多重要的实际问题。例如：噪音产生或控制方面的问题；如“音爆”反射，通过剪切、冲击或快速移动的表面的反射或传播产生和放大声音。另一个例子发生在肾结石的治疗（碎石术）中，使用集中的弱电击。很明显，对弱震聚焦技术细节的把握对于减少这种技术产生的“附带”损伤——活组织破坏——是非常重要的。＊＊＊

9311438 Rosales ` ` Research in Nonlinear Waves for Dispersive and Hyperbolic Systems of Equations is proposed. The following projects will be conducted: (1) "Focusing and Reflection of Weak Shocks and the von Neumann Paradox": These problems and related ones arising in the theory of Weakly Nonlinear Geometrical Optics ( WNGO ) will be examined (2) "Resonant Periodic Waves in Hyperbolic Systems" and (3) "Resonant Oscillations in Closed Tubes": Using WNGO, some interesting and surprising solutions for the equations of Gas Dynamics, and other hyperbolic systems, will be obtained and studied (4) "Long Weakly nonlinear Surface Waves in a Ferro-Fluid": Simplified equations for surface waves in a Ferro--Fluid will be derived (using Long Wave Asymptotics) and studied (5) "Wave Modulation Breakdown in Nonlinear Dispersive Systems": The behavior waves exhibit, after current theories to model nonlinear dispersive wave propagation breakdown and cease to apply, will be studied (6) "Square Wave Detonations": Using a simplified model, the reasons and the mechanisms that cause the "Catastrophic Instabilities" in the Square Wave Theory for Detonation Waves will be studied. This proposal deals with research in Nonlinear Waves. Waves are important in many fields, such as Acoustics, Optics, Gas Dynamics, Combustion Theory, Ocean Dynamics, Solid Mechanics, etc. Many advances have been made in Non-linear Wave Theo ry in the last few decades, but many puzzles remain. An example of the Wave Theory problems in this proposal is the "von Neumann Paradox" of Weak Shock Reflection and Focusing. This problem has been the object of much study since John von Neumann pointed it out, but remains open. The P.I. and a graduate student advanced significantly in prior research and are very close to a complete resolution. A resolution of the Paradox will have considerable scientific significance and will shed light into many important practical problems. Example: issues in noise generation or control; such as "sonic boom" reflection, generation and amplification of sound by reflection or transmission from shear, shocks or rapidly moving surfaces. Another example occurs in the treatment of kidney stones (Lithotripsy) by use of focused weak shocks. It seems clear that a firm grasp of the details of Weak Shock Focusing should be very important in reducing the amount of "collateral" damage --living tissue destruction-- produced by this technique. ***