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Nonlinearization of the Lax system for AKNS hierarchy. (English) Zbl 0714.58026

Summary: The Lax system for the AKNS vector field is nonlinearized and becomes naturally compatible under the constraint induced by a relation \((q,r)=f(\psi)\) between reflectionless potentials and the eigenfunctions of the Zakharov-Shabat eigenvalue problem (ZS). The spatial part (ZS) is nonlinearized as a completely integrable system in the Liouville sense with the Hamiltonian: \[ H=<iZ\psi_ 1,\psi_ 2>+(1/2)<\psi_ 1,\psi_ 1\times \psi_ 2,\psi_ 2> \] in the symplectic manifold \(({\mathbb{R}}^{2N},d\psi_ 1\wedge d\psi_ 2)\), whose solution variety \({\mathcal N}\) is an invariant set of the S-flow defined by the nonlinearized time part. Moreover, f maps \({\mathcal N}\) into the solution variety of a stationary AKNS equation, and maps the S-flow on \({\mathcal N}\) into the AKNS-flow on f(\({\mathcal N})\).

MSC:

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.)
35Q58 Other completely integrable PDE (MSC2000)
35Q51 Soliton equations
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