# zbMATH — the first resource for mathematics

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.)
##### Keywords:
Korteweg-de Vries hierarchy; complete integrability
Full Text:
##### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.