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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.

MSC:
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.)
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References:
[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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