Mathematical Sciences: Nonlinear Hyperbolic Waves

[404152] Hunter提出的工作目标是在连续介质力学和经典场论中理解非线性双曲波的行为。主要的方法是用奇异摄动法推导出能以最简单形式捕捉基本非线性现象的简化方程。然后用现代解析和数值方法研究这些简化方程。具体问题包括弱激波反射中的规则反射向不规则反射的转变、随机介质中的非线性波以及经典场论中的非线性波。许多种类的波——如声波、弹性波和磁流体动力波——都可以用双曲偏微分方程来描述。当振幅足够大时，这些波是非线性的。非线性的影响往往是戏剧性的，并且在性质上不同于线性理论的预测。例如，非线性会导致冲击的形成，或者有时会导致其他类型的奇点，以及各种非线性不稳定性。这个建议涉及到各种各样的涉及非线性双曲波的物理问题。这些问题包括激波的反射、大振幅声波在湍流中的传播以及大尺寸液晶定向场中定向波的分析。

9404152 Hunter The goal of the proposed work is to understand the behavior of nonlinear hyperbolic waves in continuum mechanics and classical field theory. The main approach is to use singular perturbation methods to derive reduced equations which capture essential nonlinear phenomena in their simplest form. These reduced equations are then studied using modern analytical and numerical methods. Specific problems include the transition from regular to irregular reflection in weak shock reflection, nonlinear waves in random media, and nonlinear waves in classical field theories. Many kinds of waves --- such as sound waves, elastic waves, and magnetohydrodynamic waves --- are described by hyperbolic partial differential equations. At large enough amplitudes, these waves are nonlinear. The effects of nonlinearity are often dramatic and are qualitatively different from the predictions of a linear theory. For example, nonlinearity leads to the formation of shocks, or sometimes to other kinds of singularities, and to various kinds of nonlinear instabilities. This proposal concerns a wide variety of physical problems involving nonlinear hyperbolic waves. The problems include the reflection of shock waves, the propagation of large amplitude sound waves through turbulent fluids, and the analysis of orientation waves in a massive liquid crystal director field.