Geometrical shock dynamics, formation of singularities and topological bifurcations of converging shock fronts

We are concerned with singularities of the shock fronts of converging perturbed shock waves. Our considerations are based on Whitham's theory\footnote{G.B. Whitham, {\it Linear and Nonlinear Waves}, (John Wiley \& Sons, 1974).} of geometrical shock dynamics. The recently developed method of local analysis\footnote{J. Eggers and M. A. Fontelos, {\it Panoramas et Synth\`eses}, {\bf 38}, 69 (2013).} is applied in order to determine generic singularities. In this case the solutions of partial differential equations describing the geometry of the shock fronts are presented as families of smooth maps with state variables and the set of control parameters dependent on Mach number, time and initial conditions. The space of control parameters of the singularities is analysed, the unfoldings describing the deformations of the canonical germs of shock front singularities are found and corresponding bifurcation diagrams are constructed.