Dupin hypersurfaces and integrable Hamiltonian systems of hydrodynamic type which do not possess Riemann invariants
Systems of hydrodynamic type \[ u^i_t = v^i_j (u)u^i_x, \quad i,j = 1, \dots, n \tag{1} \] are considered. The system is Hamiltonian if the matrix \(v^i_j (u)\) coincides with the Hessian \[ v^i_j (u) = \varepsilon^i {\partial^2 h(u) \over \partial u_i \partial u_j}. \] The system is called weakly nonlinear if each eigenvalue \(\lambda^i (u)\) of the matrix \(v^i_j (u)\) is constant along the corresponding eigenfoliation. It is known that all diagonalizable Hamiltonian systems (i.e. the systems which can be transformed into the form with diagonal matrix \(v^i_j)\) are integrable. The author classifies the nondiagonalizable Hamiltonian systems. The classification is based on a close relationship between Hamiltonian systems of hydrodynamic type and hypersurfaces in a pseudo-euclidean space. Let \(n\) and \(r\) be the unit normal and the radius-vector of a hypersurface \(M^n\) in pseudo-euclidean space \(E^{n + 1}\). The equation (1) is rewritten in the form \(n_t = r_x\) with \(v^i_j = (w^{-1})^i_j\), where \(w^i_j (u)\) is the Weingarten operator of \(M^n\). The classification of the 3-dimensional Dupin surfaces (principal curvatures are constant along the corresponding curvature surfaces) in \(E^4\) [\textit{U. Pinkall}, Math. Ann. 270, 427-440 (1985; Zbl 0538.53004)] is used to prove the main result of the paper: Theorem. A \(3 \times 3\) nondiagonalizable Hamiltonian system is integrable if and only if it is weakly nonlinear. All \(3 \times 3\) weakly nonlinear nondiagonalizable Hamiltonian systems are equivalent under the reciprocal transformations.
Hamiltonian systems of hydrodynamic type
scientific article; zbMATH DE number 778516
scientific article