Systems of three differential equations of hydrodynamic type with hexagonal 3-web of characteristics on the solutions

Abstract

In this article we obtain complete description of the class of hyperbolic systems of three differential equations of hydrodynamic type, for which the characteristics form hexa- gonal 3-web on each solution. For systems in Riemann invariants the hexagonality condition turns out to be equivalent to the weak nonlinearity and semi-Hamiltonianness, which enables us to integrate these systems by the generalized method of hodograph (i). However, in the nondiagonalizable case the hexagonality conditions are apparently in no way connected with semi-Hamilto~iaru~ess and describe the class of the systems that are reduced by the transfor- mations with respect to the solution to constant eigenvalues. The geometry of the charac- teristics on the solutions of quasilinear hyperbolic systems of differential equations has been studied earlier in (2) by the method of Cartan exterior forms. In (3) a classification of the particular solutions of equations for an incompressible ideal liquid and also for simultaneous nonstationa~y flow of a polytropic gas, on which the characteristics form hexa- gonal 3-web, has been carried out. It4is convenient to reduce the system, under consideration, of hydrodynamic type u i -