Geodesic
Hamiltonian Flows on Euler-Type Equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 144 (2005) no. 1, pp. 83-93 Cet article a éte moissonné depuis la source Math-Net.Ru

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We analyze properties of Hamiltonian symmetry flows on hyperbolic Euler–Liouville-type equations $\mathcal E_{EL}'$. We obtain the description of their Noether symmetries assigned to the integrals of these equations. The integrals provide Miura transformations from $\mathcal E_{EL}'$ to the multicomponent wave equations $\mathcal E$. Using these substitutions, we generate an infinite-Hamiltonian commutative subalgebra $\mathfrak A$ of local Noether symmetry flows on $\mathcal E$ proliferated by weakly nonlocal recursion operators. We demonstrate that the correspondence between the Magri schemes for $\mathfrak A$ and for the induced “modified” Hamiltonian flows $\mathfrak B\subset\operatorname{sym}\mathcal E_{EL}'$ is such that these properties are transferred to $\mathfrak B$ and the recursions for $\mathcal E_{EL}'$ are factored. We consider two examples associated with the two-dimensional Toda lattice.
Keywords: two-dimensional Toda lattice, KdV equation, commutative hierarchies.
Mots-clés : Boussinesq equation, Miura transformation 
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A. V. Kiselev. Hamiltonian Flows on Euler-Type Equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 144 (2005) no. 1, pp. 83-93. http://geodesic.mathdoc.fr/item/TMF_2005_144_1_a8/

[1] A. V. Zhiber, V. V. Sokolov, UMN, 56:1 (2001), 63–106 | DOI | MR | Zbl

[2] F. Magri, J. Math. Phys., 19:5 (1978), 1156–1162 | DOI | MR | Zbl

[3] R. M. Miura, J. Math. Phys., 9 (1968), 1202–1204 | DOI | MR | Zbl

[4] A. V. Kiselev, A. V. Ovchinnikov, J. Dynam. Control Systems, 10:3 (2004), 431–451 | DOI | MR | Zbl

[5] A. N. Leznov, V. G. Smirnov, A. B. Shabat, TMF, 51:1 (1982), 10–21 | MR | Zbl

[6] A. P. Fordy, J. Gibbons, J. Math. Phys., 21:10 (1980), 2508–2510 ; 22:6 (1981), 1170–1175 | DOI | MR | Zbl | DOI | MR | Zbl

[7] M. V. Pavlov, Fund. prikl matem., 10:1 (2004), 175–182 | MR | Zbl

[8] Y. Nutku, M. V. Pavlov, J. Math. Phys., 43:3 (2002), 1441–1459 | DOI | MR | Zbl

[9] A. V. Bocharov, A. M. Verbovetskii, A. M. Vinogradov i dr., Simmetrii i zakony sokhraneniya uravnenii matematicheskoi fiziki, eds. A. M. Vinogradov i I. S. Krasilschik, Faktorial, M., 1997

[10] A. G. Meshkov, TMF, 63:3 (1985), 323–332 | MR | Zbl

[11] G. Barnich, F. Brandt, M. Henneaux, Commun. Math. Phys., 174 (1995), 57–92 ; E-print hep-th/9405109 | DOI | MR

[12] A. V. Kiselev, Acta Appl. Math., 83:1–2 (2004), 175–182 | DOI | MR | Zbl

[13] C. S. Gardner, J. Math. Phys., 12:8 (1971), 1548–1551 | DOI | MR | Zbl

[14] P. Kersten, I. Krasil'shchik, A. Verbovetsky, J. Geom. Phys., 50:1–4 (2004), 273–302 | DOI | MR | Zbl

[15] A. B. Shabat, R. I. Yamilov, Eksponentsialnye sistemy tipa I i matritsy Kartana, Preprint, Bashkir. filial AN SSSR, Ufa, 1981

[16] D. K. Demskoi, S. Ya. Startsev, Fund. prikl. matem., 10:1 (2004), 29–37 | MR | Zbl

[17] P. Olver, Prilozheniya grupp Li k differentsialnym uravneniyam, Mir, M., 1989 | MR | Zbl

[18] E. Getzler, Duke Math. J., 111 (2002), 535–560 | DOI | MR | Zbl