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A unified Hamiltonian system on polynomial bundles and the structure of stationary problems. (Russian. English summary) Zbl 0535.58017

The paper studies the Hamiltonian structure on the phase space of solutions for nonlinear equations, which are integrable within Zakharov-Shabat scheme with \(U\) and \(V\) operators given in the form \(U(x,\lambda)=\sum^{N}_{i=0}u_ i(x)\lambda^ i\). The author defines the universal phase space \({\mathcal O}^ N\) and the set of equations \(\partial_{t_ n}U=\partial_ xV_ n+[V_ n,U]\), \(n=1,...,N\), on it. Then the universality (in the sense of independence from \(N\)) of the Hamiltonian structure is established.
The structure under consideration is restricted to the stationary variety \(\partial_ xV_ n+[V_ n,U]=0\) and the connection between Hamiltonian formalisms for stationary and nonstationary problems is investigated. A symplectic form on the space of stationary solutions, previously defined in terms of the Legendre-Ostrogradsky transformation is shown to coincide with the Kirillov form on the corresponding orbit.

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
35Q99 Partial differential equations of mathematical physics and other areas of application
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