Semi-hyperbolic patches of solutions of the pressure gradient system

We construct patches of self-similar solutions, in which one family out of two nonlinear families of characteristics starts on sonic curves and ends on transonic shock waves, to the two-dimensional pressure gradient system. This type of solutions is common in the solutions of two-dimensional Riemann problems, as seen from numerical experiments. They are not determined by the hyperbolic domain of determinacy in the traditional sense. They are middle-way between the fully hyperbolic (supersonic) and elliptic region, which we call semi-hyperbolic or partially hyperbolic. Our intention is to use the patches as building tiles to construct global solutions to general Riemann problems.