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An influential result of F. John states that no genuinely non-linear strictlyhyperbolic quasi-linear first order system of partial differential equations intwo variables has a global $C^2$-solution for small enough initial data.Inspired by recent work of D. Christodoulou, we revisit John's original proofand extract a more precise description of the behaviour of solutions at thetime of shock. We show that John's singular first order quantity, whenexpressed in characteristic coordinates, remains bounded until the final time,which is then characterised by an inverse density of characteristics tending tozero in one point. Moreover, we study the derivatives of second order, showingagain their boundedness when expressed in appropriate coordinates. We alsorecover John's upper bound for the time of shock formation and complement itwith a lower bound. Finally, we apply these results to electromagnetic planewaves in a crystal with no magnetic properties and cubic electric non-linearityin the energy density, assuming no dispersion.

On the formation of shocks of electromagnetic plane waves in non-linear crystals