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The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. II. (English) Zbl 0698.70013
Summary: [For part I see the first author, J. Math. Phys. 30, No.2, 330-338 (1989; Zbl 0678.70015).]
An isospectral problem with four potentials is discussed. The corresponding hierarchy of nonlinear evolution equations is derived. It is shown that the AKNS, Levi, D-AKNS hierarchies and a new one are reductions of the above hierarchy. In each case the relevant Hamiltonian form is established by making use of the trace identity.

70H05 Hamilton’s equations
35Q99 Partial differential equations of mathematical physics and other areas of application
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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.)
Full Text: DOI
[1] Tu, G. Z., Constrained Formal Variational Calculus and Its Applications to Soliton Equations,Scientia Sinica,24 A, (1986), 138–148. · Zbl 0614.49030
[2] Tu, G. Z., On Generalized Hamiltonian Structures of Infinite-Dimensional Integrable Systems,Adv. Sci. China, ser. Math.,2 (1987), 45–72.
[3] Tu, G. Z., A Trace Identity, A Powerful Tool for constructing the Hamiltonian Structure of Integrable Systems,J. Math.Phys.,30 (1989) (to Appear). · Zbl 0678.70015
[4] Tu, G. Z., A New Hierarchy of Integrable Systems and Its Hamiltonian Structures,Scientia Sinica,31:12 (1988), 28–39.
[5] Tu, G. Z., On Liouville Integrability of Zero Curvature Equations and the Yang Hierarchy (to appear). · Zbl 0697.58025
[6] Tu, G. Z., A Simple Approach to Hamiltonian Structure of Soliton Equations II,Sci. Exploration,2 (1982), 85–92.
[7] Levi, D., Neugebauer, G. and Meinel, R., A New Nonlinear Schrodinger Equation, Its Hierarchy andN-Soliton Solutions,Phys. Lett.,102A (1984), 1–6.
[8] Giachetti, R. and Johnson, R., A Hamiltonian Structure From Gauge Transformations of the Zakharov-Shabat System,Phys. Lett.,102A (1984), 81–82.
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