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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 $\psi\sb x=U\psi$, where $U=U(u,\lambda)$ is a matrix depending on $u=(u\sb 1(x,t),...,u\sb p(x,t))$ and a spectral parameter $\lambda$, we can relate it to a hierarchy of t-evolution equations $\psi\sb{t\sb n}=V\sp{(n)}\psi$ such that the compatibility condition of the above two equations, which reads $U\sb{t\sb n}-V\sb x\sp{(n)}+UV-VU=0,$ represents a meaningful hierarchy of nonlinear evolution equations on u: $u\sb{t\sb n}=f\sb n(u)$. It is also known that in most cases the above hierarchy of equations can be cast to their Hamiltonian form: $u\sb{t\sb n}=f\sb n(u)=J\delta H\sb n/\delta u$, where J is a Hamiltonian operator. The main result of this paper is a formula for the Poisson bracket: $\{H(\lambda),H(\mu)\}=(d/dx)f(\lambda,\mu)$, where $H(\lambda)=\sum H\sb n\lambda\sp{-n}$ and the function f($\lambda$,$\mu)$ is explicitly constructed. As an immediate consequence of this formula, it is shown that $\{H\sb m,H\sb n\}$ are total derivatives with respect to x. It means that the flows $J\delta H\sb n/\delta u$ commute to each other, thus it proves generally that the hierarchy of equations $u\sb{t\sb n}=J\delta H\sb n/\delta u$ are Liouville integrable. Moreover in a subsequent paper we show that the explicit form of f($\lambda$,$\mu)$ can be used to construct effectively the Hamiltonians for the stationary Lagrangian equations $\delta H\sb n/\delta u=0$.
Reviewer: Tu Guizhang

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
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