On integrability of 3×3 semi-Hamiltonian hydrodynamic type systems uit = vij( u) ujx which do not possess Riemann invariants

Abstract

We give a complete classification of the integrable 3×3 semi-Hamiltonian hydrodynamic type systems which do not possess Riemann invariants. The “integrability” is understood as the existence of at least one higher integral. All these equations are transformable to the system of resonant 3-wave interaction by appropriate differential substitution and hence are actually integrable via inverse scattering transform. Considerations are based on a convenient coordinate-free approach to nondiagonalizable systems.