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Parametric representation of the finite-band solution of the Heisenberg equation. (English) Zbl 0961.35507

Summary: The Heisenberg magnetic equation (HM) is studied and it is proved that every finite-band potential \(u\), which is the solution of the stationary HM, has a parametric representation \(u=f(\psi)\), where \(\psi\) is a solution of a finite-dimensional integrable system obtained through the nonlinearization of the associated eigenvalue problem. The approach is valid for other soliton hierarchies. A relation with a \((2+1)\)-dimensional HM is also presented.

MSC:

35Q58 Other completely integrable PDE (MSC2000)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
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