Articles

MÖBIUS-TODA HIERARCHY AND ITS INTEGRABILITY

  • Chuanzhong LI
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  • Department of Mathematics, Ningbo University, Ningbo 315211, China
Chuanzhong LI,E-mail:lichuanzhong@nbu.edu.cn

Received date: 2016-02-01

  Revised date: 2016-09-24

  Online published: 2017-08-25

Supported by

This work was supported by the National Natural Science Foundation of China (11571192), the Natural Science Foundation of Ningbo (2015A610157) and K. C. Wong Magna Fund in Ningbo University.

Abstract

In this paper, we construct a new integrable equation called Möbius-Toda equation which is a generalization of q-Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of the Möbius-Toda equation and a whole integrable Möbius-Toda hierarchy are also constructed. To show the integrability, the bi-Hamiltonian structure and tau symmetry of the Möbius-Toda hierarchy are given and this leads to the existence of the tau function.

Cite this article

Chuanzhong LI . MÖBIUS-TODA HIERARCHY AND ITS INTEGRABILITY[J]. Acta mathematica scientia, Series B, 2017 , 37(4) : 1151 -1161 . DOI: 10.1016/S0252-9602(17)30063-2

References

[1] Toda M. Vibration of a chain with nonlinear interaction. J Phys Soc Jpn, 1967, 22:431-436
[2] Toda M. Nonlinear Waves and Solitons. Dordrecht, Holland:Kluwer Academic Publishers, 1989
[3] Ueno K, Takasaki K. Toda lattice hierarchy//Group Representations and Systems of Differential Equations (Tokyo, 1982). Adv Stud Pure Math, 4. Amsterdam:North-Holland, 1984:1-95
[4] Witten E. Two-dimensional gravity and intersection theory on moduli space. Surveys Differ Geom, 1991, 1:243-310
[5] Dubrovin B A. Geometry of 2D topological field theories//Integrable Systems and Quantum Groups (Montecatini Terme, 1993). Lecture Notes in Math, 1620. Berlin:Springer, 1996:120-348
[6] Carlet G, Dubrovin B, Zhang Y. The extended Toda hierarchy. Moscow Math J, 2004, 4:313-332
[7] Carlet G. The extended bigraded Toda hierarchy. J Phys A, 2006, 39:9411-9435
[8] 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, 2010, 51:043514
[9] Li C Z. Solutions of bigraded Toda hierarchy. J Phys A, 2011, 44:255201
[10] Li C Z, He J S. Dispersionless bigraded Toda hierarchy and its additional symmetry. Reviews Math Phys, 2012, 24:1230003
[11] Li C Z, He J S, Su Y C. Block type symmetry of bigraded Toda hierarchy. J Math Phys, 2012, 53:013517
[12] Li C Z, He J S, The extended ZN-Toda hierarchy. Theor Math Phys, 2015, 185:1614-1635
[13] Meng A, Li C Z, Huang S. Integrability on generalized q-Toda equation and hierarchy. J Nonlinear Math Phys, 2014, 21:429-441
[14] Li C Z, Sato theory on the q-Toda hierarchy and its extension. Chaos Solitons Fract, 2015, 76:10-23
[15] Kac V G, van de Leur J W. The n-component KP hierarchy and representation theory. J Math Phys, 2003, 44:3245
[16] Mas J, Seco M. The algebra of q-pseudodifferential symbols and the q-WKP(n) algebra. J Math Phys, 1996, 37:6510-6529
[17] Tu M H. q-deformed KP hierarchy:its additional symmetries and infinitesimal Bäcklund transformations. Lett Math Phys, 1999, 49:95-103
[18] Lin R, Liu X, Zeng Y. A new extended q-deformed KP hierarchy. J Nonlinear Math Phys, 2008, 15:333-347
[19] Iliev P, Tau function solutions to a q-deformation of the KP hierarchy. Lett Math Phys, 1998, 44:187-200
[20] He J S, Li Y H, Cheng Y. q-deformed KP hierarchy and q-deformed constrained KP hierarchy. SIGMA, 2006, 2:060
[21] Tian K L, He J S, Su Y C, Cheng Y. String equations of the q-KP hierarchy. Chin Ann Math, Series B, 2011, 32:895-904
[22] Li C Z, Li T T. Virasoro symmetry of the (r, m)-component q-constrained KP hierarchy. submitted
[23] Tsuboi Z, Kuniba A. Solutions of a discretized Toda field equation for Dr from analytic Bethe ansatz. J Phys A, 1996, 29:7785-7796
[24] Silindir B. Soliton solutions of q-Toda lattice by Hirota direct method. Adv Differ Equ, 2012, 2012:121
[25] Sato M. Soliton Equations as Dynamical Systems on a Infinite Dimensional Grassmann Manifolds. RIMS Kokyuroku, 1982, 439:30-46
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