The Geometry of Hamiltonian Systems of Hydrodynamic Type. The Generalized Hodograph Method

Geometry of 2D topological field theories

“…of (6.33a) In this case from the results of Tsarev [136] it follows completeness of the family of the conservation laws (6.26) for any of the systems in the hierarchy (6.25).…”

“…For massive F robenius manifolds these form a dense subset in the space of all solutions of (6.25) (see [136,46]). Formally they can be obtained from the solution (6.42) by a shift of the arguments T ;p .…”

WDVV and DZM

“…both ρ andρ change). We note also that, according to [23,24], the theory of Combescure transformations coincides with the theory of integrable diagonal systems of hydrodynamic type.…”

Differential geometry of hydrodynamic Vlasov equations

“…With these hypotheses, the system can then be integrated by the generalized hodograph transformation ( [3]). We remark that the semiHamiltonian property is automatically satisfied for a Hamiltonian system with Dubrovin-Novikov Hamiltonian structure, and that the conditions for the system to be respectively diagonalizable, or semi-Hamiltonian, can be written invariantly; each corresponds to the vanishing of some tensor ( [4], [1]).…”

“…An obvious problem related with systems of hydrodynamic type (1) is to determine whether such a system is integrable, in the sense that it admits infinitely many conserved densities and commuting flows; in [3], Tsarev proved that this is true if the system is hyperbolic and can be written in diagonal form R i t = λ i (R)R i x , where R i are called the Riemann invariants and where the λ i (called the characteristic velocities) satisfy the semi-Hamiltonian condition…”