Mathematical Sciences: Nonlinear Hyperbolic Waves

小行星9404152 这项工作的目标是了解连续介质力学和经典场论中非线性双曲波的行为。主要的方法是使用奇异摄动法推导出约化方程，以最简单的形式捕捉基本的非线性现象。这些减少方程，然后研究使用现代分析和数值方法。具体问题包括弱激波反射中规则反射到不规则反射的过渡，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学波，都可以用双曲型偏微分方程来描述。在足够大的振幅下，这些波是非线性的。非线性的影响往往是戏剧性的，与线性理论的预测有质的不同。例如，非线性会导致激波的形成，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物理问题，包括非线性双曲波。这些问题包括冲击波的反射，大振幅声波通过湍流流体的传播，以及大量液晶指向矢场中的取向波的分析。

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.