# CAIMS Prize Award

Department of Mathematical and Statistical Sciences, University of Alberta

Supervisor: B. Moodie

Thesis title: Geometrical Optics For Nonlinear Conservation Laws And Shock Wave Dynamics

Abstract:

Weakly nonlinear hyperbolic waves arising from the action of initial or boundary disturbances in systems of one dimensional conservation laws are considered. The study undertaken is divided into two parts. The first part includes a relatively complete single-wave-mode geometric optics theory, in which we investigate weakly nonlinear hyperbolic waves subject to small-amplitude, high-frequency boudary disturbances of single-wave-mode type. By introducing a nonlinear phase from the outset, an asymptotic solution to the signalling problem is constructed. A rational scheme is designed to adjust the small-amplitude to high-frequency relation according to the order of the local linear degeneracy. The transition process from smooth wave breaking to the generation of shock waves is carefully studied via bifurcation analysis. We show that wave-breaking will lead to the generation of entropy admissible shock waves. Shock fitting and tracking are also accomplished. As a prototypical example, this process is demonstrated in a transparent fashion for scalar conservation laws in the large.

The second part is devoted to a non-resonant two-wave interaction theory which generalizes a characteristic method originally introduced by C.C.Lin. Both initial and signalling problems are investigated for systems of conservation laws through the deployment of asymptotic analysis. We apply this theory to compute the interaction and propagation of two weak sound waves in one dimensional gas dynamics. The theory is also applied to study an interesting problem arising from the context of geophysical fluid dynamics, that is, nonlinear Kelvin waves confined to a channel.