LAPLACE TRANSFORMATIONS OF HYDRODYNAMIC TYPE SYSTEMS IN RIEMANN INVARIANTS

The conserved densities of hydrodynamic-type systems in Riemann invari ants satisfy a system of linear second-order partial differential equa tions. For linear systems of this type, Darboux introduced Laplace tra nsformations, which generalize the classical transformations of a seco nd-order scalar equation. It is demonstrated that the Laplace transfor mations can be pulled back to transformations of the corresponding hyd rodynamic-type systems. We discuss finite families of hydrodynamic-typ e systems that are closed under the entire set of Laplace transformati ons. For 3 x 3 systems in Riemann invariants, a complete description o f closed quadruples is proposed. These quadruples appear to be related to a special quadratic reduction of the (2 + 1)-dimensional 3-wave sy stem.