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On Liouville integrability of zero-curvature equations and the Yang hierarchy. (English) Zbl 0697.58025
Nowadays it is known that for a properly chosen isospectral problem ψ x =Uψ, where U=U(u,λ) is a matrix depending on u=(u 1 (x,t),···,u p (x,t)) and a spectral parameter λ, we can relate it to a hierarchy of t-evolution equations ψ t n =V (n) ψ such that the compatibility condition of the above two equations, which reads U t n -V x (n) +UV-VU=0, represents a meaningful hierarchy of nonlinear evolution equations on u: u t n =f n (u). It is also known that in most cases the above hierarchy of equations can be cast to their Hamiltonian form: u t n =f n (u)=JδH n /δu, where J is a Hamiltonian operator. The main result of this paper is a formula for the Poisson bracket: {H(λ),H(μ)}=(d/dx)f(λ,μ), where H(λ)=H n λ -n and the function f(λ,μ) is explicitly constructed. As an immediate consequence of this formula, it is shown that {H m ,H n } are total derivatives with respect to x. It means that the flows JδH n /δu commute to each other, thus it proves generally that the hierarchy of equations u t n =JδH n /δu are Liouville integrable. Moreover in a subsequent paper we show that the explicit form of f(λ,μ) can be used to construct effectively the Hamiltonians for the stationary Lagrangian equations δH n /δu=0.
Reviewer: Tu Guizhang

MSC:
37J35Completely integrable systems, topological structure of phase space, integration methods
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
35Q99PDE of mathematical physics and other areas