Mathematical Sciences: Asymptotic Analysis of Nonlinear Hyperbolic Waves

本文研究了非线性双曲 偏微分方程及其在波中的应用 在各种物理环境中传播。 该分析乃 基于复杂方程组的系统简化 通过渐近方程将模型方程简化为更简单的模型方程。 模型方程的研究，使用显式的组合 解，扰动方法，定性分析，和 数值计算 本研究属于一般应用领域 数学 扰动在其中传播的物理系统 有限速度通常用双曲型偏微分方程来模拟。 微分方程 实例包括可压缩气流， 弹性力学、磁流体力学和广义相对论。 当 扰动的速度取决于扰动的强度 扰动，方程是非线性的。 一位杰出 非线性双曲波的性质是形成 震荡 冲击波具有重要的实际意义 (e.g.在音爆或非手术性肾损伤中 石头），并提出了许多具有挑战性的数学问题。

This research addresses the analysis of nonlinear hyperbolic partial differential equations, with applications to wave propagation in a variety of physical contexts. The analysis is based on the systematic reduction of complex systems of equations to simpler model equations by means of asymptotic equations. The model equations are studied using a combination of explicit solutions, perturbation methods, qualitative analysis, and numerical calculations. The research lies in the general area of applied mathematics. Physical systems in which disturbances propagate at finite speeds are often modelled by hyperbolic partial differential equations. Examples include compressible gas flows, elasticity, magnetohydrodynamics, and general relativity. When the speed of the disturbance depends on the strength of the disturbance, the equations are nonlinear. An outstanding property of nonlinear hyperbolic waves is the formation of shocks. Shock waves are of significant practical importance (e.g. in sonic booms or the nonsurgical destruction of kidney stones) and pose many challenging mathematical problems.