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.
