Abstract
In this paper, we construct a new even constrained B(C)-type Toda hierarchy and derive its B(C)-type Block-type additional symmetry. Also we generalize the B(C)-type Toda hierarchy to the N-component B(C)-type Toda hierarchy which is proved to have symmetries of a coupled \(\bigotimes ^NQT_+ \) algebra (N-fold direct product of the positive half of the quantum torus algebra QT).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
Álvarez-Fernández, C., Prieto, U.F., Mañas, M.: Multiple orthogonal polynomials of mixed type: Gauss–Borel factorization and the multi-component 2D Toda hierarchy. Adv. Math. 227, 1451–1525 (2011)
Carlet, G.: The extended bigraded Toda hierarchy. J. Phys. A Math. Theor. 39, 9411–9435 (2006)
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, 405205–405220 (2013)
Carlet, G., Dubrovin, B., Zhang, Y.: The extended Toda hierarchy. Mosc. Math. J. 4, 313–332 (2004)
Cheng, J.P., Tian, K.L., He, J.S.: The additional symmetries for the BTL and CTL hierarchies. J. Math. Phys. 51, 053515 (2011)
Dijkgraaf, R., Witten, E.: Mean field theory, topological field theory, and multimatrix models. Nucl. Phys. B 342, 486–522 (1990)
Li, C.Z.: Solutions of bigraded Toda hierarchy. J. Phys. A Math. Theor. 44, 255201 (2011). arXiv:1011.4684
Li, C.Z., He, J.S.: Dispersionless bigraded Toda hierarchy and its additional symmetry. Rev. Math. Phys. 24, 1230003 (2012)
Li, C.Z., He, J.S.: On the extended multi-component Toda hierarchy. Math. Phys. Anal. Geom. 17, 377–407 (2014)
Li, C.Z., He, J.S.: The extended \(Z_N\)-Toda hierarchy. Theor. Math. Phys. 185, 1614–1635 (2015)
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)
Li, C.Z., He, J.S., Su, Y.C.: Block type symmetry of bigraded Toda hierarchy. J. Math. Phys. 53, 013517 (2012)
Mañas, M., Alonso, L.M.: The multicomponent 2D Toda hierarchy: dispersionless limit. Inverse Probl. 25, 11 (2009)
Mañas, M., Alonso, M.L., Álvarez-Fernández, C.: The multicomponent 2D Toda hierarchy: discrete flows and string equations. Inverse Probl. 25, 065007 (2009)
Toda, M.: Wave propagation in anharmonic lattices. J. Phys. Soc. Jpn. 23, 501–506 (1967)
Ueno, K., Takasaki, K.: Toda lattice hierarchy, In Group representations and systems of differential equations (Tokyo, 1982), 1–95, Adv. Stud. Pure Math. 4, North-Holland, Amsterdam (1984)
Acknowledgements
This work is supported by the National Natural Science Foundation of China under Grant No. 11571192 and K. C. Wong Magna Fund in Ningbo University.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Anthony Bloch.
Rights and permissions
About this article
Cite this article
Wang, N., Li, C. Quantum Torus Algebras and B(C)-Type Toda Systems. J Nonlinear Sci 27, 1957–1970 (2017). https://doi.org/10.1007/s00332-017-9392-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-017-9392-3
Keywords
- B(C)-type Toda hierarchy
- Additional symmetries
- Multicomponent B(C)-type Toda hierarchy
- Quantum torus algebra