Development of singularities in Riemann Invariants

Shocks form in finite time in systems of quasilinear hyperbolic equations in one space variable which are genuinely nonlinear. The authors write down a simple geometric construction for systems of two equations, and use it to obtain a priori estimates for the growth of the derivatives. They also find realistic bounds on the maximum and minimum time of existence of smooth solutions for large amplitude waves in a model system of an unusual type