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On the Extended Multi-component Toda Hierarchy

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Abstract

The extended flow equations of the multi-component Toda hierarchy are constructed. We give the Hirota bilinear equations and tau function of this new extended multi-component Toda hierarchy(EMTH). Because of logarithmic terms, some extended Vertex operators are constructed in generalized Hirota bilinear equations which might be useful in topological field theory and Gromov-Witten theory. Meanwhile the Darboux transformation and bi-hamiltonian structure of this hierarchy are given. From the hamiltonian tau symmetry, we give another different tau function of this hierarchy with some unknown mysterious connections with the one defined from the point of wave functions.

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References

  1. Toda, M.: Vibration of a chain with nonlinear interaction. J. Phys. Soc. Jpn. 22, 431–436 (1967)

    Article  ADS  Google Scholar 

  2. Toda, M.: Nonlinear waves and solitons. Dordrecht, Kluwer Academic Publishers (1989)

    MATH  Google Scholar 

  3. Ueno, K., Takasaki, K.: Toda lattice hierarchy. In: Group Representations and Systems of Differential Equations (Tokyo, 1982), Adv. Stud. Pure Math., pp. 1–95, p 4. North-Holland, Amsterdam (1984)

    Google Scholar 

  4. Witten, E.: Two-dimensional gravity and intersection theory on moduli space. Surv Differ Geom 1, 243–310 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dubrovin, B.A.: Geometry of 2D topological field theories. In: Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., 1620, pp. 120–348. Springer, Berlin (1996)

    Book  Google Scholar 

  6. Carlet, G., Dubrovin, B., Zhang, Y.: The Extended Toda Hierarchy. Moscow Math. J. 4, 313–332 (2004)

    MathSciNet  MATH  Google Scholar 

  7. Milanov, T.: Hirota quadratic equations for the extended Toda hierarchy. Duke Math. J. 138, 161–178 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carlet, G.: The extended bigraded Toda hierarchy. J. Phys. A 39, 9411–9435 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Li, C.Z., He, J.S., Wu, K., Cheng, Y.: Tau function and Hirota bilinear equations for the extended bigraded Toda Hierarchy. J. Math. Phys. 51, 043514 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Aoyama, S., Kodama, Y.: Topological Landau-Ginzburg theory with a rational potential and the dispersionless KP hierarchy. Commun. Math. Phy. 182, 185–219 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Milanov, T., Tseng, H.H.: The spaces of Laurent polynomials, \(\mathbb {P}^{1}\)-orbifolds, and integrable hierarchies. J. für die reine und Angew. Math. 622, 189–235 (2008)

    MATH  Google Scholar 

  12. Li, C.Z.: Solutions of bigraded Toda hierarchy. J. Phys. A 44, 255201 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Li, C.Z., He, J.S.: Dispersionless bigraded Toda hierarchy and its additional symmetry. Rev. Math. Phys. 24, 1230003 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, C.Z., He, J.S., Su, Y.C.: Block type symmetry of bigraded Toda hierarchy. J. Math. Phys. 53, 013517 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Milanov, T., Shen, Y.F., Tseng, H.H.: Gromov-Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda Hierarchies, arXiv: 1401.5778

  16. Kac, V.G., van de Leur, J.W.: The n-component KP hierarchy and representation theory. J. Math. Phys. 44, 3245 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Adler, M., van Moerbeke, P., Vanhaecke, P.: Moment matrices and multi-component KP, with applications to random matrix theory. Commun. Math. Phys. 286, 1–38 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Mañas, M., Martínez Alonso, L.: The multicomponent 2D Toda hierarchy: dispersionless limit. Inverse Probl. 25, 11 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Mañas, M., Martínez Alonso, L., Álvarez Fernández, C.: The multicomponent 2D Toda hierarchy: discrete flows and string equations. Inverse Probl 25, 065007 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Álvarez Fernández, C., Fidalgo Prieto, U., Mañas, M.: The multicomponent 2D Toda hierarchy: generalized matrix orthogonal polynomials, multiple orthogonal polynomials and Riemann–Hilbert problems. Inverse Probl. 26, 055009 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Álvarez Fernández, C., Fidalgo Prieto, U., Mañas, M.: Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchy. Adv. Math. 227, 1451–1525 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Carlet, G., van de Leur, J. : Hirota equations for the extended bigraded Toda hierarchy and the total descendent potential of \(\mathbb {P}^{1}\) orbifolds. J. Phys. A: Math. Theor. 46, 405205 (2013). arxiv: 1304.1632

    Article  MathSciNet  MATH  Google Scholar 

  23. He, J.S., Zhang, L., Cheng, Y., Li, Y.S.: Determinant representation of Darboux transformation for the AKNS system. Sci. China A 12, 1867–78 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, C.Z., He, J.S., Porsezian, K.: Rogue waves of the Hirota and the Maxwell-Bloch equation. Phys. Rev. E 87, 012913 (2013)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Ningbo University, Ningbo, 315211, China

    Chuanzhong Li & Jingsong He

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  1. Chuanzhong Li
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  2. Jingsong He
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Correspondence to Jingsong He.

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Li, C., He, J. On the Extended Multi-component Toda Hierarchy. Math Phys Anal Geom 17, 377–407 (2014). https://doi.org/10.1007/s11040-014-9162-5

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  • DOI: https://doi.org/10.1007/s11040-014-9162-5

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