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Fan, Engui

Integrable evolution systems based on Gerdjikov-Ivanov equations, bi-Hamiltonian structure, finite-dimensional integrable systems and \(N\)-fold Darboux transformation. (English) Zbl 0986.37059

J. Math. Phys. 41, No. 11, 7769-7782 (2000).
Author’s abstract: A spectral problem and the associated Gerdjikov-Ivanov (GI) hierarchy of nonlinear evolution equations is presented. As a reduction, the well-known GI equation of derivative nonlinear Schrödinger equations is obtained. It is shown that the GI hierarchy is integrable in a Liouville sense and possesses bi-Hamiltonian structure. Moreover, the spectral problem can be nonlinearized as a finite dimensional completely integrable system under the Bargmann constraint between the potentials and the eigenfunctions. In particular, an explicit \(N\)-fold Darboux transformation for the GI equation is constructed with the help of a gauge transformation of spectral problems and a reduction technique. Some explicit solitonlike solutions of the GI equation are given by applying its Darboux transformation.
Reviewer: Georgi E.Karadzhov (Sofia)

Cited in 80 Documents

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q58 Other completely integrable PDE (MSC2000)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests

Keywords:

nonlinear Schrödinger equations; solitons; Darboux transformation
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