Reduction procedures for hyperbolic equations: applications to nonlinear wave problems.

Partial differential equations (PDEs) play a key role in the description of a widerange of complex phenomena such as the propagation of heat or sound, fluid flow,elasticity, electrostatics, electrodynamic and so on. For this reason, determining solutions of PDEs is a great challenge in applied mathematics and mechanics.Motivated by this viewpoint, the aim of this thesis is to briefly review some of themost useful procedures for solving PDEs and to develop new approaches to constructexact solutions for this kind of models. Both the cases of partial differentialequations of higher order and of systems of first order are treated.Hyperbolic PDEs of higher order are studied in the framework of the intermediateintegrals method: a reduction procedure to simplify the problem of solving a secondorder equation to the one of studying a first order equation is developed andan algorithm that under appropriate circustances permits to determine the generalintegral of linear equations is proposed.On the other hand, hyperbolic systems are approached through the method of differential constraints: the well known Riemann problems (RP), Riemann problemswith structure (RPS) and the generalised Riemann problems (GRP) are considered.At the state of art, a general theory for solving this type of problems for non homogeneous systems does not exist: in this thesis, we provide results concerning thesolution of a Riemann problem for a traffic flow model and of GRP and RPS for thenon homogeneous p-system.