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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.

70H05Hamilton’s equations
35Q99PDE of mathematical physics and other areas
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
37J35Completely integrable systems, topological structure of phase space, integration methods
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
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.