A reduction method to quasilinear hyperbolic systems of multicomponent field PDEs with application to wave interaction

A reduction method is worked out for determining a class of exact solutions with inherent wave features to quasilinear hyperbolic homogeneous systems of N>2 first-order autonomous PDEs. A crucial point of the present approach is that in the process the original set of field equations induces the hyperbolicity of an auxiliary 2×2 subsystem and connection between the respective characteristic velocities can be established. The integration of this auxiliary subsystem via the hodograph method and through the use of the Riemann invariants provides the searched solutions to the full governing system. These solutions also represent invariant solutions associated with groups of translation of space/time coordinates and involving arbitrary functions that can be used for studying non-linear wave interaction. Within such a theoretical framework the two-dimensional motion of an adiabatic fluid is considered. For appropriate model pressure-entropy-density laws, we determine a solution to the governing system of equations which describes in the 2+1 space two non-linear waves which were initiated as plane waves, interact strongly on colliding but emerge with unaffected profile from the interaction region. These model material laws include the classical pressure-entropy-density law which is usually adopted for a polytropic fluid.