LAPLACE TRANSFORMATIONS OF HYDRODYNAMIC-TYPE SYSTEMS IN RIEMANN INVARIANTS - PERIODIC SEQUENCES

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 tran sformations, generalizing the classical transformations in the scalar case. It is demonstrated that Laplace transformations can be pulled ba ck to the transformations of the corresponding hydrodynamic-type syste ms. We discuss periodic Laplace sequences of 2 x 2 hydrodynamic-type s ystems with emphasis on the simplest nontrivial case of period 2. For 3 x 3 systems in Riemann invariants a complete discription of closed q uadruples is proposed. They turn out to be related to a special quadra tic reduction of the (2+1)-dimensional 3-wave system which can be redu ced to a triple of pairwise commuting Monge-Ampere equations. In terms of the Lame and rotation coefficients Laplace transformations have a natural interpretation as the symmetries of the Dirac operator, associ ated with the (2 + 1)-dimensional n-wave system. The 2-component Lapla ce transformations can also be interpreted as the symmetries of the (2 + 1)-dimensional integrable equations of Davey-Stewartson type. Lapla ce transformations of hydrodynamic-type systems originate front a cano nical geometric correspondence between systems of conservation laws an d line congruences in projective space.