On the matrix Hopf equation and integrable Hamiltonian systems of hydrodynamic type, which do not possess Riemann invariants

Abstract

We propose a complete description of the integrable Hamiltonian 3 x 3 systems of hydrodynamic type, which do not possess Riemann invariants: the system of this class is integrable if and only if it is weakly nonlinear (linearly degenerate). Any 3 x 3 weakly nonlinear nondiagonalizable Hamiltonian system can be transformed to the matrix Hopf equation U t =( U 2) x , where U is a 3 x 3 symmetric matrix subject to the constraints tr U=const, tr U 2=const. The matrix Hopf equation is equivalent to the system of resonant three-wave interaction and hence is integrable via the inverse scattering transform. We formulate several conjectures and unsolved problems concerning the structure and general properties of the integrable Hamiltonian systems of hydrodynamic type.