Geodesic
The extended $Z_N$-Toda hierarchy
Teoretičeskaâ i matematičeskaâ fizika, Tome 185 (2015) no. 2, pp. 289-312 Cet article a éte moissonné depuis la source Math-Net.Ru

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We construct the extended flow equations of a new $Z_N$-Toda hierarchy taking values in a commutative subalgebra $Z_N$ of $gl(N,\mathbb C)$. We give the Hirota bilinear equations and tau function of this new extended $Z_N$-Toda hierarchy. Taking the presence of logarithmic terms into account, we construct some extended vertex operators in generalized Hirota bilinear equations, which might be useful in topological field theory and the Gromov–Witten theory. We present the Darboux transformations and bi-Hamiltonian structure of this hierarchy. Using Hamiltonian tau-symmetry, we obtain another tau function of this hierarchy with some unknown mysterious relation to the tau function derived using the Sato theory.
Keywords: extended $Z_N$-Toda hierarchy, Hirota quadratic equation, bi-Hamiltonian structure.
Mots-clés : Darboux transformation 
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Chuanzhong Li; Jingsong He. The extended $Z_N$-Toda hierarchy. Teoretičeskaâ i matematičeskaâ fizika, Tome 185 (2015) no. 2, pp. 289-312. http://geodesic.mathdoc.fr/item/TMF_2015_185_2_a3/

[1] M. Toda, J. Phys. Soc. Japan, 22:2 (1967), 431–436 | DOI

[2] M. Toda, Nonlinear Waves and Solitons, Mathematics and its Applications (Japanese Series), 5, Kluwer, Dordrecht, 1989 | MR | Zbl

[3] K. Ueno, K. Takasaki, “Toda lattice hierarchy”, Group Representations and Systems of Differential Equations (Tokyo, 1982), Advanced Studies in Pure Mathematics, 4, ed. K. Okamoto, North-Holland, Amsterdam, 1982, 1–95 | MR

[4] E. Witten, “Two-dimensional gravity and intersection theory on moduli space”, Surveys in Differential Geometry (Harvard University, April 27–29, 1990), v. I, Proceedings of the Conference on Geometry and Topology, eds. S. T. Yau, H. B. Lawson, Jr., AMS, Providence, RI, 1991, 243–310 | DOI | MR | Zbl

[5] B. A. Dubrovin, “Geometry of 2D topological field theories”, Integrable Systems and Quantum Groups (Montecatini Terme, Italy, June 14–22, 1993), Lecture Notes in Mathematics, 1620, eds. M. Francaviglia, S. Greco, Springer, Berlin, 1996, 120–348 | DOI | MR | Zbl

[6] G. Carlet, B. Dubrovin, Y. Zhang, Mosc. Math. J., 4:2 (2004), 313–332 | MR | Zbl

[7] T. Milanov, Duke Math. J., 138:1 (2007), 161–178 | DOI | MR | Zbl

[8] G. Carlet, J. Phys. A: Math. Gen., 39:30 (2006), 9411–9435, arXiv: math-ph/0604024 | DOI | MR | Zbl

[9] C. Z. Li, J. S. He, K. Wu, Y. Cheng, J. Math. Phys., 51:4 (2010), 043514, 32 pp., arXiv: 1003.5987 | DOI | MR | Zbl

[10] S. Aoyama, Y. Kodama, Commun. Math. Phys., 182:1 (1996), 185–219, arXiv: hep-th/9505122 | DOI | MR | Zbl

[11] T. Milanov, H. H. Tseng, J. Reine Angew. Math., 2008:622 (2008), 189–235 | DOI | MR | Zbl

[12] C. Z. Li, J. Phys. A: Math. Theor., 44:25 (2011), 255201, 29 pp., arXiv: 1011.4684 | DOI | MR | Zbl

[13] C. Z. Li, J. S. He, Rev. Math. Phys., 24:7 (2012), 1230003, 34 pp. | DOI | MR | Zbl

[14] C. Z. Li, J. S. He, Y. C. Su, J. Math. Phys., 53:1 (2012), 013517, 24 pp. | DOI | MR | Zbl

[15] V. G. Kac, J. W. van de Leur, J. Math. Phys., 44:8 (2003), 3245–3293, arXiv: hep-th/9308137 | DOI | MR | Zbl

[16] M. Adler, P. van Moerbeke, P. Vanhaecke, Commun. Math. Phys., 286:1 (2009), 1–38 | DOI | MR | Zbl

[17] M. Mañas, L. Martínez Alonso, Inverse Problems, 25:11 (2009), 115020, 22 pp. | DOI | MR | Zbl

[18] M. Mañas, L. Martínez Alonso, C. Álvarez Fernández, Inverse Problems, 25:6 (2009), 065007, 31 pp., arXiv: 0809.2720 | DOI | MR | Zbl

[19] C. Álvarez Fernández, U. Fidalgo Prieto, M. Mañas, Inverse Problems, 26:5 (2010), 055009, 15 pp., arXiv: 0911.0941 | DOI | MR

[20] C. Álvarez Fernández, U. Fidalgo Prieto, M. Mañas, Adv. Math., 227:4 (2011), 1451–1525 | DOI | MR

[21] G. Carlet, J. van de Leur, J. Phys. A: Math. Theor., 46:40 (2013), 405205, 16 pp. | DOI | MR | Zbl

[22] C. Z. Li, J. S. He, Math. Phys.: Anal. Geom., 17:3–4 (2014), 377–407, arXiv: 1410.3657 | DOI | MR | Zbl

[23] D. F. Zuo, Local matrix generalizations of $W$-algebras, arXiv: 1401.2216

[24] C. Z. Li, J. S. He, K. Porsezian, Phys. Rev. E, 87:1 (2013), 012913, 13 pp., arXiv: 1205.1191 | DOI

[25] J. S. He, L. Zhang, Y. Cheng, Y. S. Li, Sci. China Ser. A, 49:12 (2006), 1867–1878 | DOI | MR | Zbl

[26] G. Carlet, Extended Toda hierarchy and its Hamiltonian structure, PhD Thesis, SISSA, Trieste, 2003