On Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type. (English. Russian original) Zbl 0605.35075
Sov. Math., Dokl. 31, 488-491 (1985); translation from Dokl. Akad. Nauk SSSR 282, 534-537 (1985).
The author gives the full description of the integrals of hydrodynamic type \(I=\int P(u)dx\) for the Hamiltonian systems \(u^ i_ t=v^ i_ j(u)u^ j_ x\) \((i,j=1,...,n)\) which arise in hydrodynamic and average theory. Using the generalized hodograph method (an analogue of the method of inverse problem for averaged equations) he presents an explicit procedure for integrating an arbitrary diagonal Hamiltonian system.
Reviewer: V.Vasil’ev
MSC:
35Q99 | Partial differential equations of mathematical physics and other areas of application |
35F20 | Nonlinear first-order PDEs |
37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |