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An extension of a theorem on gradients of conserved densities of integrable systems. (English) Zbl 0718.58042
The author discusses an isospectral problem, using an extension of the theorem of Boiti-Pempinelli-Tu concerning the gradient of conserved densities of integrable systems. Let (1) \(\psi_ z=U\psi\) be an isospectral problem where \(U=U(u,\lambda)\) is an \(N\times N\) matrix depending on the spectral parameter \(\lambda\) and the potential \(u=(u_ 1,...,u_ p)\). A matrix \(Y=(y_{ij})\) with \(y_{ij}=\psi_ i/\psi_ j\) is constructed. Then set \(H=(UY)_ D=diag[h_ 1,...,h_ N]\), where \(h_ i=(UY)_{ii}\), and \(V\equiv \delta H/\delta U^ T=\nabla H\) the gradient of H with \(U^ T\) the transpose of U. It is well known that \(d\tilde H/dt=0\) for (2) \(\tilde H=\int^{\infty}_{-\infty}tr(CH)dx\) where \(C=diag(c_ 1,...,c_ N)\) for arbitrary constants \(c_ 1,...,c_ N.\)
The author proves: Theorem 1. Let \(\tilde H\) be given as above. Then one has (3) (\(\nabla \tilde H)_ z=[U,\nabla \tilde H]\), where \(\nabla \tilde H\) is the gradient of \(\tilde H\) defined by \(\delta\) \(\tilde H=\int^{\infty}_{-\infty}<\nabla \tilde H,\delta U>dx\) [see D. H. Sattinger, Stud. Appl. Math. 72, 65-86 (1985; Zbl 0584.58022)];
Theorem 2. Let the conserved density h be given by \(\delta h/\delta u_ i=<V,\partial U/\partial u_ i>\). Then one has also the equation (3).
For the isospectral problem, the author obtains:
Theorem 3. The gradient of eigenvalues of the isospectral problem (1) satisfies the stationary zero-curvature equation \(V_ z=[U,V]\).

37A30 Ergodic theorems, spectral theory, Markov operators
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35P99 Spectral theory and eigenvalue problems for partial differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems