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Action-angle variables for the Gel’fand-Dikii flows. (English) Zbl 0754.35134
The action-angle variables for the Gel’fand-Dikii flows [see I. M. Gel’fand and L. A. Dikii, Funct. Anal. Appl. 10 (1976), 259-273 (1977; Zbl 0356.35072)], which generalize the Korteweg-de Vries hierarchy, are considered, and complete integrability in a strong sense is proved.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
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] R. Beals and R. Coifman, Linear spectral problems, nonlinear equations, and the \(\mathop \partial \limits^ - \) , Inverse Problems5, 87-130 (1989). · Zbl 0685.35080
[2] R. Beals, P. Deift, and C. Tomei,Direct and Inverse Scattering on the Line, Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, R.I.28 (1989). · Zbl 0679.34018
[3] R. Beals and D. H. Sattinger,On the complete integrability of completely integrable systems, Comm. in Math. Phys.138, 404-436 (1991). · Zbl 0727.58022
[4] R. Beals and D. H. Sattinger,Complete integrability of ?completely integrable? systems, Proc. Conference on Inverse Scattering Problems, Amherst, Mass., in Contemporary Mathematics (1990). · Zbl 0743.35065
[5] C. S. Gardner,Korteweg-deVries equation and generalizations, IV?The Korteweg-deVries equations as a Hamiltonian system, J. Math. Phys.12, 1548-1551 (1971). · Zbl 0283.35021
[6] I. M. Gel’fand and L. A. Dikii,Fractional powers of operators and Hamiltonian systems, Funct. Anal. Appl.10, 259-273 (1976). · Zbl 0356.35072
[7] D. W. McLaughlin,Four examples of the inverse method as a canonical transformation, J. Math. Phys.16, 96-99 (1975). · Zbl 0292.35018
[8] D. H. Sattinger,Hamiltonian hierarchies on semi-simple Lie algebras, Stud. Appl. Maths.72, 65-86 (1985). · Zbl 0584.58022
[9] V. E. Zakharov and L. D. Faddeev,The Korteweg-deVries equation as a completely integrable Hamiltonian system. Funct. Anal. Appl.5, 280-287 (1971). · Zbl 0257.35074
[10] V. E. Zakharov and S. V. Manakov,On the complete integrability of the nonlinear Schrödinger equation, Teor. Mat. Fyz.19, 332-343 (1974). · Zbl 0293.35025
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