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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 $U=U\left(u,\lambda \right)$ is an $N×N$ matrix depending on the spectral parameter $\lambda$ and the potential $u=\left({u}_{1},···,{u}_{p}\right)$. A matrix $Y=\left({y}_{ij}\right)$ with ${y}_{ij}={\psi }_{i}/{\psi }_{j}$ is constructed. Then set $H={\left(UY\right)}_{D}=diag\left[{h}_{1},···,{h}_{N}\right]$, where ${h}_{i}={\left(UY\right)}_{ii}$, 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 $d\stackrel{˜}{H}/dt=0$ for (2) $\stackrel{˜}{H}={\int }_{-\infty }^{\infty }tr\left(CH\right)dx$ where $C=diag\left({c}_{1},···,{c}_{N}\right)$ for arbitrary constants ${c}_{1},···,{c}_{N}·$

The author proves: Theorem 1. Let $\stackrel{˜}{H}$ be given as above. Then one has (3) ($\nabla \stackrel{˜}{H}{\right)}_{z}=\left[U,\nabla \stackrel{˜}{H}\right]$, where $\nabla \stackrel{˜}{H}$ is the gradient of $\stackrel{˜}{H}$ defined by $\delta$ $\stackrel{˜}{H}={\int }_{-\infty }^{\infty }<\nabla \stackrel{˜}{H},\delta U>dx$ [see D. H. Sattinger, Stud. Appl. Math. 72, 65-86 (1985; Zbl 0584.58022)];

Theorem 2. Let the conserved density h be given by $\delta h/\delta {u}_{i}=$. 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 ${V}_{z}=\left[U,V\right]$.

##### MSC:
 37A30 Ergodic theorems, spectral theory, Markov operators 37J35 Completely integrable systems, topological structure of phase space, integration methods 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 35P99 Spectral theory and eigenvalue problems for PD operators 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems