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

MSC:
 37J35 Completely integrable systems, topological structure of phase space, integration methods 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems 35Q99 PDE of mathematical physics and other areas