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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 $$V_ z=[U,V]$$.

##### MSC:
 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
Zbl 0584.58022