Mathematical Sciences: Nonlinear Hyperbolic and Dispersive Waves

点击翻译按钮获取中文摘要

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. ***