Mathematical Sciences: Asymptotic Analysis of Nonlinear Hyperbolic Waves

本研究解决了非线性双曲型偏微分方程的分析，并应用于各种物理环境下的波传播。分析的基础是利用渐近方程将复杂方程组系统化简为更简单的模型方程。模型方程采用显式解、摄动法、定性分析和数值计算相结合的方法进行研究。本研究属于应用数学的一般范畴。扰动以有限速度传播的物理系统通常用双曲型偏微分方程来模拟。例子包括可压缩气体流动、弹性、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时，方程是非线性的。非线性双曲波的一个突出特性是激波的形成。冲击波具有重要的实际意义（例如，在音爆或肾结石的非手术破坏中），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.