Publication: The multicomponent 2D Toda hierarchy: dispersionless limit
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2009-11
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IOP Publishing
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Abstract
The factorization problem of the multi-component 2D Toda hierarchy is used to analyze the dispersionless limit of this hierarchy. A dispersive version of the Whitham hierarchy defined in terms of scalar Lax and Orlov-Schulman operators is introduced and the corresponding additional symmetries and string equations are discussed. Then, it is shown how KP and Toda pictures of the dispersionless Whitham hierarchy emerge in the dispersionless limit. Moreover, the additional symmetries and string equations for the dispersive Whitham hierarchy are studied in this limit.
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©IOP Publishing.
The authors wish to thank the Spanish Ministerio de Ciencia e Innovaciòn, research project FIS2008- 00200, and acknowledge the support received from the European Science Foundation (ESF) and the activity entitled Methods of Integrable Systems, Geometry, Applied Mathematics (MISGAM). MM wish to thank Prof. van Moerbeke and Prof. Dubrovin for their warm hospitality, acknowledge economical support from MISGAM and SISSA and reckons different conversations with P. van Moerbeke, T. Grava, G. Carlet and M. Caffasso. MM also acknowledges to Prof. Liu for his invitation to visit the China Mining and Technology University at Beijing.
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[1] M. Mañas, L. Martínez Alonso, and C. Alvarez-Ferández, The multi-component 2D Toda Hierarchy: discrete flows and string equations arXiv:0809.2720. To be published in Inverse Problems.
[2] K. Ueno and K. Takasaki, Adv. Stud. Pure Math. 4 (1984) 1.
[3] P. Di Francesco, P. Ginsparg and Z. Zinn- Justin, Phys. Rept. 254 (1995) 1.
[4] B. Eynard, An Introduction to Random Matrices, lectures given at Saclay, October 2000, http://wwwspht.cea.fr/articles/t01/014/.
[5] L. Bonora and C. S. Xiong, Phys. Lett. B347 (1995) 41.
[6] K. Takasaki, Commun. Math. Phys. 181 (1996) 131.
[7] K. Takasaki and T. Takebe, Rev. Math. Phys. 7 (1995) 743.
[8] P. B. Wiegmann and A. Zabrodin, Comm. Math. Phys. 213 (2000) 523.
M. Mineev- Weinstein, P.B. Wiegmann, and A. Zabrodin, Phys. Rev. Lett. 84 (2000) 5106
I. M. Krichever, M. Mineev-Weinstein, P. B. Wiegmann, and A. Zabrodin, Physica D198 (2004) 1.
I. M. Krichever, A. Marshakov and A. Zabrodin, Comm. Math. Phys. 259 (2005) 1.
M. Mañas, L. Martínez Alonso, and E. Medina, J. Phys. A: Math. Gen. 35 (2002) 401.
F. Guil, M. Mañas, and L. Martínez Alonso, J. Phys. A: Math. Gen. 36 (2003) 4047.
F. Guil, M. Mañas, and L. Martínez Alonso, J. Phys. A: Math. Gen. 36 (2003) 6457.
L. Martínez Alonso and M. Mañas, J. Math. Phys. 44 (2003) 3294.
M. Mañas, J. Phys. A: Math. Gen. 37 (2004) 9195.
M. Mañas, J. Phys. A: Math. Gen. 37 (2004) 11191.
[9] E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, J. Phys. Soc. Japan 40 (1981) 3806.
M. J. Bergvelt and A. P. E. ten Kroode, Pacific J. Math. 171 23-88 (1995).
M. Mañas, L. Martínez Alonso, and E. Medina, J. Phys.A: Math. Gen. 33 (2000) 2871.
M. Mañas, L. Martínez Alonso, and E. Medina, J. Phys.A: Math. Gen. 33 (2000) 7181.
V. G. Kac and J. W. van de Leur, J. Math. Phys. 44 (2003) 3245.
[10] M. Bertola, B. Eynard and J. Harnad, Comm. Math. Phys. 229 (2002) 73.
[11] M. Bertola, B. Eynard and J. Harnad, Comm. Math. Phys. 243 (2003) 193.
[12] A. B. J. Kuijlaars and K. T-R McLaughlin , J. Comput. Appl. Math. 178 (2005) 313.
[13] M. Adler and P. Van Moerbeke, Comm. Pure and Appl. Math. J. 50 (1997) 241 ; Ann. Math. 149 (1999) 921 .
[14] M. Adler, P. van Moerbeke, and P. Vanhaecke, Commun. Math. Phys. 286 (2009) 1. M.Adler. J. Delépine and P. van Moerbeke, Comm. Pure and Appl. Math. 62 (2009) 334.
[15] P. M. Bleher and A. B. J. Kuijlaars, Int. Math. Research Notices 2004 (2004) 109; Comm. Math. Phys. 252 (2004) 43; Comm. Math. Phys. 270 (2007) 481.
[16] L. Martínez Alonso, E. Medina, Multiple orthogonal polynomials, string equations and the large-n limit arXiv:0812.3817. To be published in J. Phys. A.
[17] W. Van Assche, J. S. Geronimo, and A. B. J. Kuijlaars, Riemann-Hilbert problems for multiple orthogonal polynomials, In: Special Functions 2000: Current Perspectives and Future Directions (J. Bustoz et al., eds.), Kluwer, Dordrecht, (2001) 23.
[18] E. Daems and A. B. J. Kuijlaars, J. of Approx. Theory 146 (2007) 91.
[19] E. Daems, Asymptotics for non- intersecting Brownian motions using multiple orthogonal polynomials, Ph.D. thesis, K.U.Leuven, 2006, URL http://hdl.handle.net/1979/324.
[20] A.S. Fokas, A.R. Its and A.V. Kitaev, Commun. Math. Phys. 147 (1992) 395.
[21] P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert approach, Courant Lecture Notes in Mathematics Vol. 3, Amer. Math. Soc., Providence R.I. 1999.
[22] P. Deift and X. Zhou, Ann. Math. 2 (1993) 137 .
[23] A. Yu Orlov and E. I. Schulman, Lett. Math. Phys. 12 (1986) 171.
[24] I. M. Krichever, Comm. Pure. Appl. Math. 47 (1994) 437.
[25] K. Takasaki, “Dispersionless integrable hierarchies revisited”, talk delivered at SISSA at september 2005 (MISGAM program).
[26] K. Takasaki and T. Takebe, Physica D235
(2007) 109.
[27] L. Martínez Alonso, E. Medina and M. Mañas, J. Math. Phys. 47 (2006) 083512. M. Mañas, E. Medina and L. Martínez Alonso, J. Phys.A: Math. Gen. 39 (2006) 2349.