No Access

GEODESIC FLOW ON EXTENDED BOTT–VIRASORO GROUP AND GENERALIZED TWO-COMPONENT PEAKON TYPE DUAL SYSTEMS

PARTHA GUHA

Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, D-04103 Leipzig, Germany

S. N. Bose National Center for Basic Sciences, JD Block, Sector-3, Salt Lake, Calcutta-700098, India

https://doi.org/10.1142/S0129055X08003523Cited by:6 (Source: Crossref)

Abstract

This paper discusses an algorithmic way of constructing integrable evolution equations based on Lie algebraic structure. We derive, in a pedagogical style, a large class of two-component peakon type dual systems from their two-component soliton equations counter part. We study the essential aspects of Hamiltonian flows on coadjoint orbits of the centrally extended semidirect product group to give a systematic derivation of the dual counter parts of various two-component of integrable systems, viz., the dispersive water wave equation, the Kaup–Boussinesq system and the Broer–Kaup system, using moment of inertia operators method and the (frozen) Lie–Poisson structure. This paper essentially gives Lie algebraic explanation of Olver–Rosenau's paper [31].

Keywords:

AMSC: 53A07, 53B50

References

E. Arbarelloet al., Comm. Math. Phys. 117, 1 (1988), DOI: 10.1007/BF01228409. Web of Science, ADS, Google Scholar
M. Antonowicz and A. Fordy, Phys. D 28, 345 (1987), DOI: 10.1016/0167-2789(87)90023-6. Web of Science, Google Scholar
O. Babelon, D. Bernard and M. Talon, Introduction to Classical Integrable Systems, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, 2003) p. 602. Google Scholar
R. Camassa and D. D. Holm, Phys. Rev. Lett. 71, 1661 (1993), DOI: 10.1103/PhysRevLett.71.1661. Web of Science, ADS, Google Scholar
R. Camassa, D. D. Holm and J. M. Hyman, Adv. Appl. Mech. 31, 1 (1994), DOI: 10.1016/S0065-2156(08)70254-0. Google Scholar
A. Constantin and B. Kolev, Phys. Lett. A 350(1–2), 75 (2006), DOI: 10.1016/j.physleta.2005.10.011. Web of Science, ADS, Google Scholar
A. Constantin and B. Kolev, J. Nonlinear Sci. 16(2), 109 (2006), DOI: 10.1007/s00332-005-0707-4. Web of Science, ADS, Google Scholar
D. D. Holm, J. E. Marsden and T. S. Ratiu, Adv. Math. 137, 1 (1998), DOI: 10.1006/aima.1998.1721. Web of Science, Google Scholar
M. Chen, S.-Q. Liu and Y. Zhang, A 2-component generalization of the Camassa–Holm equation and its solutions, preprint, nlin.SI/0501028 (2005) . Google Scholar
G. Falqui, On a Camassa–Holm type equation with two dependent variables, preprint, nlin.SI/0505059 (2005) . Google Scholar
A. S. Fokas, Phys. D 87, 145 (1995), DOI: 10.1016/0167-2789(95)00133-O. Web of Science, Google Scholar
A. S. Fokas and B. Fuchssteiner, Lett. Nuovo Cimento 28(2), 299 (1980), DOI: 10.1007/BF02798794. Web of Science, Google Scholar
B. Fuchssteiner and A. S. Fokas, Phys. D 4, 47 (1981), DOI: 10.1016/0167-2789(81)90004-X. Web of Science, Google Scholar
A. S. Fokas, P. J. Olver and P. Rosenau, Algebraic Aspects of Integrable System: In Memory of Irene Dorfman, Progress in Nonlinear Equations 26, eds. A. S. Fokas and I. M. Gel'fand (Birkhauser, 1996) pp. 93–101. Google Scholar
B. Fuchssteiner, Progr. Theoret. Phys. 65(3), 861 (1981), DOI: 10.1143/PTP.65.861. Web of Science, ADS, Google Scholar
B. Fuchssteiner, Phys. D 95, 229 (1996), DOI: 10.1016/0167-2789(96)00048-6. Web of Science, Google Scholar
I. M. Gelfand and I. Ya. Dorfman, Funct. Anal. Appl. 14(3), 71 (1980). Web of Science, Google Scholar
I. M. Gelfand, I. M. Graev and A. M. Vershik, Representations of Lie groups and Lie Algebras (Akad. Kiad, Budapest, 1985) pp. 121–179. Google Scholar
P. Guha, Lett. Math. Phys. 52, 311 (2000), DOI: 10.1023/A:1007660018819. Web of Science, Google Scholar
P. Guha, J. Math. Anal. Appl. 310, 45 (2005), DOI: 10.1016/j.jmaa.2004.12.060. Web of Science, Google Scholar
P. Guha, J. Appl. Anal. 11, 261 (2005). Web of Science, Google Scholar
P. Guha and P. J. Olver, SIGMA Symmetry Integrability Geom. Methods Appl. 2, 9 (2006). Google Scholar
M. Ito, Phys. Lett. A 91, 335 (1982), DOI: 10.1016/0375-9601(82)90426-1. Web of Science, ADS, Google Scholar
B. Khesin and G. Misiolek, Adv. Math. 176(1), 116 (2003), DOI: 10.1016/S0001-8708(02)00063-4. Web of Science, Google Scholar
B. Kolev, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 365, 2333 (2007), DOI: 10.1098/rsta.2007.2012. Web of Science, ADS, Google Scholar
B. A. Kupershmidt, J. Phys. A 18(18), L1119 (1985), DOI: 10.1088/0305-4470/18/18/001. ADS, Google Scholar
S.-Q. Liu and Y. Zhang, J. Geom. Phys. 54, 427 (2005), DOI: 10.1016/j.geomphys.2004.11.003. Web of Science, ADS, Google Scholar
P. Marcel, V. Ovsienko and C. Roger, Lett. Math. Phys. 40, 31 (1997), DOI: 10.1023/A:1007310811875. Web of Science, Google Scholar
G. Misiolek, J. Geom. Phys. 24, 203 (1998). Web of Science, ADS, Google Scholar
J. Marsden, T. Ratiu and A. Weinstein, Trans. Amer. Math. Soc. 281(1), 147 (1984), DOI: 10.2307/1999527. Web of Science, Google Scholar
P. Olver and P. Rosenau, Phys. Rev. E 53, 1900 (1996), DOI: 10.1103/PhysRevE.53.1900. Web of Science, Google Scholar
V. Yu. Ovsienko and B. A. Khesin, Funct. Anal. Appl. 21, 329 (1987), DOI: 10.1007/BF01077813. Web of Science, Google Scholar
P. Rosenau, Phys. Rev. Lett. 73, 1737 (1994), DOI: 10.1103/PhysRevLett.73.1737. Web of Science, ADS, Google Scholar