Mathematical Sciences: Nonlinear Hyperbolic Waves

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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.