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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 PDE of mathematical physics and other areas 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems 37J35 Completely integrable systems, topological structure of phase space, integration methods 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
##### Keywords:
isospectral problem
##### References:
 [1] Tu, G. Z., Constrained Formal Variational Calculus and Its Applications to Soliton Equations,Scientia Sinica,24 A, (1986), 138–148. [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). [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). [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.