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Isospectral Hamiltonian flows in finite and infinite dimensions. I: Generalized Moser systems and moment maps into loop algebras. (English) Zbl 0659.58022
The authors provide a systematic link between finite dimensional integrable systems, flows in loop algebras and “integrable” partial differential equations through the use of moment maps. In particular they use the theory of Adler-Kostant-Symes on Hamiltonian systems on coadjoint orbits to produce a large class of commuting flows of isospectral type that are generalizations of the rank two perturbations considered by J. Moser [Differential geometry, Proc. int. Chern Symp., Berkeley 1979, 147-188 (1980; Zbl 0455.58018)].
Reviewer: H.Knörrer

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.)
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
37C10 Dynamics induced by flows and semiflows
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[1] Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math.53, 249-315 (1973) · Zbl 0408.35068
[2] Ablowitz, M.J., Segur, H.: Solitons and the inverse scattering transform. SIAM Studies in Applied mathematics,4, society for industrial and applied mathematics. Philadelphis 1981 · Zbl 0472.35002
[3] Abraham, R., Marsden, J.E.: Foundations of mechanics, 2nd ed. Reading, MA: Benjamin/Cummings 1978, Chap. 4 · Zbl 0393.70001
[4] Adams, M.R., Harnad, J., Hurtubise, J.: Isospectral hamiltonian flows in finite and infinite dimensions. II. Integration of flows (preprint) (1988) · Zbl 0717.58051
[5] Adler, M.: On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-de Vries type equations. Invent. Math.50, 219-248 (1979) · Zbl 0393.35058 · doi:10.1007/BF01410079
[6] Adler, M., van Moerbeke, P.: Completely integrable systems, euclidean Lie algebras, and curves. Adv. Math.38, 267-317 (1980) · Zbl 0455.58017 · doi:10.1016/0001-8708(80)90007-9
[7] Adler, M., van Moerbeke, P.: Linearization of Hamiltonian systems, Jacobi varieties, and representation theory. Adv. Math.38, 318-379 (1980) · Zbl 0455.58010 · doi:10.1016/0001-8708(80)90008-0
[8] Date, E., Tanaka, S.: Periodic multi-soliton solutions of the Korteweg-de Vries equation and Toda lattice. Prog. Theor. Phys. [Suppl.]59, 107 (1976) · Zbl 1109.37307 · doi:10.1143/PTPS.59.107
[9] Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Proc. Jpn. Acad.57A, 342 and 387 (1981); Physica4D, 343 (1982); J. Phys. Soc. Jpn.50, 3806 and 3813 (1981); Publ. RIMS Kyoto Univ.18, 1077 (1982)
[10] Deift, P., Lund, F., Trubowitz, E.: Nonlinear wave equations and constrained harmonic motion. Commun. Math. Phys.74, 141-188 (1980) · Zbl 0435.35072 · doi:10.1007/BF01197756
[11] Dhooge, P.: Bäcklund transformations on Kac-Moody Lie algebras and integrable systems. J. Geom. Phys.1, 9-38 (1984) · Zbl 0579.58012 · doi:10.1016/0393-0440(84)90002-0
[12] Dubrovin, B.A.: Theta functions and non-linear equations. Russ. Math. Surv.36, 11-92 (1981) · Zbl 0549.58038 · doi:10.1070/RM1981v036n02ABEH002596
[13] Flaschka, H.: Towards an algebro-geometric interpretation of the Neumann system. Tohoku Math. J.36, 407-426 (1984) · Zbl 0582.35102 · doi:10.2748/tmj/1178228807
[14] Flaschka, H., Newell, A.C., Ratiu, T.: Kac-Moody Lie algebras and soliton equations. II. Lax equations associated withA 1 (1) . Physica9D, 300 (1983) · Zbl 0643.35099
[15] Kac-Moody Lie algebras and soliton equations. III. Stationary equations associated withA 1 (1) . Physica9D, 324-332 (1983)
[16] Forest, M.G., McLaughlin, D.W.: Spectral theory for the periodic sine-Gordon equation: a concrete viewpoint. J. Math. Phys.23, 1248-1277 (1982) · Zbl 0498.35072 · doi:10.1063/1.525509
[17] Gagnon, L., Harnad, J., Hurtubise, J., Winternitz, P.: Abelian integrals and the reduction method for an integrable Hamiltonian system. J. Math. Phys.26, 1605-1612 (1985) · Zbl 0597.70020 · doi:10.1063/1.526926
[18] Guillemin, V., Sternberg, S.: Symplectic techniques in physics. Cambridge: Cambridge University Press 1984 · Zbl 0576.58012
[19] Guillemin, V., Sternberg, S.: The moment map and collective motion. Ann. Phys.127, 220-253 (1980) · Zbl 0453.58015 · doi:10.1016/0003-4916(80)90155-4
[20] Guillemin, V., Sternberg, S.: On collective complete integrability according to the method of Thimm. Ergodic Dyn. Sys.3, 219-230 (1983) · Zbl 0511.58024
[21] Guillemin, V., Sternberg, S.: On the method of Symes for integrating systems of the Toda type. Lett. Math. Phys.7, 113-115 (1983) · Zbl 0521.58033 · doi:10.1007/BF00419928
[22] Harnad, J., Saint-Aubin, Y., Shnider, S.: Bäcklund transformations for nonlinear sigma models with values. In: Riemannian symmetric spaces. Commun. Math. Phys.93, 33-56 (1984) · Zbl 0549.58025 · doi:10.1007/BF01218638
[23] The soliton correlation matrix and the reduction problem for integrable systems. Commun. Math. Phys.92, 329-367 (1984)
[24] Helgason, S.: Differential geometry and symmetric spaces. New York: Academic Press 1962, Chap. IX · Zbl 0111.18101
[25] Hurtubise, J.: Rankr perturbations, algebraic curves, and ruled surfaces, preprint U.Q.A.M. (1986)
[26] Kostant, B.: The solution to a generalized Toda lattice and representation theory. Adv. Math.34, 195-338 (1979) · Zbl 0433.22008 · doi:10.1016/0001-8708(79)90057-4
[27] Krichever, I.M.: Algebraic curves and commuting matrix differential operators. Funct. Anal. Appl.10, 144-146 (1976) · Zbl 0347.35077 · doi:10.1007/BF01077946
[28] Krichever, I.M.: Methods of algebraic geometry in the theory of non-linear equations. Russ. Math. Surv.32, 6, 185-213 (1977) · Zbl 0386.35002 · doi:10.1070/RM1977v032n06ABEH003862
[29] Krichever, I.M., Novikov, S.P.: Holomorphic bundles over algebraic curves and non-linear equations. Russ. Math. Surv.35, 6, 53 (1980) · Zbl 0548.35100 · doi:10.1070/RM1980v035n06ABEH001974
[30] McKean, H.P., Trubowitz, E.: Hill’s operator and hyperelliptic function theory in the presence of infinitely many branched points. CPAM29, 143-226 (1976); Hill’s surfaces and their theta functions, Bull. AMS84, 1042-1085 (1979) · Zbl 0339.34024
[31] Mischenko, A.S., Fomenko, A.T.: Generalized Liouville method of integration of Hamiltonian systems. Funct. Anal. Appl.12, 113-121 (1978) · Zbl 0405.58028 · doi:10.1007/BF01076254
[32] Mischenko, A.S., Fomenko, A.T.: Integrability of Euler equations on semisimple Lie algebras. Sel. Math. Sov.2, 207-291 (1982) · Zbl 0517.58024
[33] van Moerbeke, P., Mumford, D.: The spectrum of difference operators and algebraic curves. Acta Math.143, 93-154 (1979) · Zbl 0502.58032 · doi:10.1007/BF02392090
[34] Moser, J.: Geometry of quadrics and spectral theory. The chern symposium, Berkeley, June 1979; p. 147-188. Berlin, Heidelberg, New York: Springer 1980
[35] Moser, J.: Various aspects of integrable Hamiltonian systems, Proc. CIME Conference, Bressanone, Italy, June 1978; Prog. Math.8. Boston: Birkhäuser 1980
[36] Mumford, D.: Tata lectures of theta. II. Prog. Math.43. Boston: Birkhäuser 1983 · Zbl 0509.14049
[37] Pressley, A., Segal, G.: Loop groups. Oxford: Oxford University Press 1986 · Zbl 0618.22011
[38] Previato, E.: Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schrödinger equation. Duke Math. J.52 (1985) · Zbl 0578.35086
[39] Ratiu, T.: The C. Neumann problem as a completely integrable system on an adjoint orbit. Trans. AMS264, 321-329 (1981); · Zbl 0475.58006
[40] The Lie algebraic interpretation of the complete integrability of the Rosochatius system. In: Mathematical methods in hydrodynamics and integrability in dynamical systems. AIP Conf. Proc.88, La Jolla, 1981
[41] Reyman, A.G., Semenov-Tian-Shansky, M.A.: Reduction of Hamiltonian systems, affine Lie algebras and Lax equations. Invent. Math.54, 81-100 (1979) · Zbl 0415.58012 · doi:10.1007/BF01391179
[42] Reyman, A.G., Semenov-Tian-Shansky, M.A.: Reduction of Hamiltonian systems, affine Lie algebras and Lax equations II. Invent. Math.63, 423-432 (1981) · Zbl 0452.58014 · doi:10.1007/BF01389063
[43] Reyman, A.G., Semenov-Tian-Shansky, M.A., Frenkel, I.B.: Graded Lie algebras and completely integrable dynamical systems. Sov. Math. Doklady20, 811-814 (1979) · Zbl 0437.58008
[44] Sato, M.: Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds RIMS Kokyuroku439, 30-46 (1981);
[45] with Sato, Y.: Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds. In: Nonlinear PDE’s in applied science U.S.-Japan Seminar, Tokyo 1982, Lax, P., Fujita, H. (eds.). Amsterdam: North-Holland 1982
[46] Segal, G., Wilson, G.: Loop groups and equations of KdV type. Publ. Math. IHES61, 6-65 (1985) · Zbl 0592.35112
[47] Schilling, R.: Trigonal curves and operator deformation theory (preprint)
[48] Symes, W.W.: Systems of Toda type, inverse spectral problems, and representation theory. Invent. Math.59, 13-59 (1980) · Zbl 0474.58009 · doi:10.1007/BF01390312
[49] Ting, A.C., Tracy, E.R., Chen, H.H., Lee, Y.C.: Reality constraints for the periodic sine-Gordon equation, Phys. Rev. A30, 3355-3358 (1984)
[50] Tracy, E.R., Chen, H.H., Lee, Y.C.: Study of quasi-periodic solutions of the nonlinear Schrödinger equation and the nonlinear modulational instability. Phys. Rev. Lett.53, 218-221 (1984) · doi:10.1103/PhysRevLett.53.218
[51] Weinstein, A.: Lectures on symplectic manifolds. CBMS Conference Series, Vol. 29. Providence, RI: Am. Math. Soc. 1977 · Zbl 0406.53031
[52] Weinstein, A.: The local structure of Poisson manifolds. J. Differ. Geom.18, 523-557 (1983) · Zbl 0524.58011
[53] Wilson, G.: Habillage et fonctions ?. C.R. Acad. Soc.299, 587-590 (1984) · Zbl 0564.35086
[54] Zakharov, V.E., Shabat, A.B.: A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I. Funct. Anal. Appl.8, 226-235 (1974) · Zbl 0303.35024 · doi:10.1007/BF01075696
[55] Zakharov, V.E., Shabat, A.B.: Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II. Funct. Anal. Appl.18, 166-174 (1979) · Zbl 0448.35090
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