## 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