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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) ψ z =Uψ be an isospectral problem where U=U(u,λ) is an N×N matrix depending on the spectral parameter λ and the potential u=(u 1 ,···,u p ). A matrix Y=(y ij ) with y ij =ψ i /ψ j is constructed. Then set H=(UY) D =diag[h 1 ,···,h N ], where h i =(UY) ii , and VδH/δU T =H the gradient of H with U T the transpose of U. It is well known that dH ˜/dt=0 for (2) H ˜= - tr(CH)dx where C=diag(c 1 ,···,c N ) for arbitrary constants c 1 ,···,c N ·

The author proves: Theorem 1. Let H ˜ be given as above. Then one has (3) (H ˜) z =[U,H ˜], where H ˜ is the gradient of H ˜ defined by δ H ˜= - <H ˜,δU>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 δh/δu i =<V,U/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].

MSC:
37A30Ergodic theorems, spectral theory, Markov operators
37J35Completely integrable systems, topological structure of phase space, integration methods
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
35P99Spectral theory and eigenvalue problems for PD operators
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems