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Hamiltonian operators and algebraic structures related to them. (English) Zbl 0437.58009


MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
53D50 Geometric quantization
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.)

Citations:

Zbl 0428.58009
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References:

[1] I. M. Gel’fand and L. A. Dikii, ”Asymptotics of the resolvent of Sturm?Liouville equations and the algebra of Korteweg?de Vries equations,” Usp. Mat. Nauk,30, No. 5, 67-100 (1975).
[2] I. M. Gel’fand and L. A. Dikii, ”The structure of a Lie algebra in the formal calculus of variations,” Funkts. Anal. Prilozhen.,10, No. 1, 18-25 (1976).
[3] I. M. Gel’fand and L. A. Dikii, ”Fractional powers of operators and Hamiltonian systems,” Funkts. Anal. Prilozhen.,10, No. 4, 13-29 (1976).
[4] I. M. Gel’fand, Yu. I. Manin, and M. A. Shubin, ”Poisson brackets and the kernel of the variational derivatives in the formal calculus of variations,” Funks. Anal. Prilozhen.,10, No. 4, 30-34 (1976). · Zbl 0534.58024
[5] B. A. Kuperschmidt and Yu. I. Manin, ”Equations of long waves with a free surface. I. Conservation laws and solutions,” Funks. Anal. Prilozhen.,11, No. 3 (1977); 31-42; ”II. Hamiltonian structure and higher equations,” Funks. Anal. Prilozhen.,12, No. 1, 25-32 (1978).
[6] M. Adler, ”On a trace functional for formal pseudodifferential operators and the symplectic structure of the Korteweg?de Vries equation,” Invent. Math.,50, No. 3, 219-248 (1979). · Zbl 0393.35058
[7] D. R. Lebedev and Yu. I. Manin, ”A Hamiltonian operator of Gel’fand-Dikii and the coadjoint representation of the Volterra group,” Preprint ITÉF, No. 155 (1978). · Zbl 0455.58012
[8] I. M. Gel’fand and L. A. Dikii, ”A family of Hamiltonian structures connected with integrable, nonlinear differential equations,” Preprint Inst. Prikl. Mat. Akad. Nauk SSSR. No. 136 (1978).
[9] M. Flato, A. Lichnerowicz, and D. Sternheimer, ”Algebras de Lie attachées a une variété canonique,” Preprint (1975).
[10] P. D. Lax, ”Almost periodic solutions of the KdV equation,” SIAM Rev.,18, No. 3, 351-375 (1976). · Zbl 0329.35015
[11] I. Ya. Dorfman, ”On the formal variational calculus in an algebra of smooth cylinder functions,” Funkts. Anal. Prilozhen.,12, No. 2, 32-39 (1978). · Zbl 0401.49023
[12] P. J. Olver, ”Evolution equations possessing infinitely many symmetries,” J. Math. Phys.,18, No. 6, 1212-1215 (1977). · Zbl 0348.35024
[13] M. Adler, ”Some algebraic relations common to a set of integrable partial and ordinary differential equations,” Preprint (1978).
[14] F. Magri, ”A simple model of the integrable Hamiltonian equation,” J. Math. Phys.,19, No. 5, 1156-1162 (1978). · Zbl 0383.35065
[15] P. P. Kulish and A. G. Reiman, ”The hierarchy of symplectic forms for the Schrödinger and Dirac equations on the line,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,77, 134-147 (1978). · Zbl 0427.35064
[16] I. M. Gel’fand and I. Ya. Dorfman, ”Integral equations of KdV-H. Dym type,” in: Modern Problems of Computational Mathematics and Mathematical Physics [in Russian], Moscow (1979).
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