The generalized hodograph method and its applications

The generalized hodograph method was introduced by S. P. Tsarev in 1985.
It can be applied to a wide class of two-dimensional quasilinear systems of first-order equations that are diagnosable, has pairwise different characteristic velocities, and also satisfy the so-called semi-Hamiltonian condition.
In this case a two-dimensional quasilinear system of first-order equations is associated to a linear system of partial differential equations with nonconstant coefficients.
In general case, even particular solutions of such linear systems are difficult to find.
The talk will discuss the case of two-dimensional quasilinear systems of first-order equations that are hydrodynamic reductions of integrable three-dimensional systems of first-order equations. In this case a complete set of particular solutions can be effectively computed.