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Dispersionless integrable equations as coisotropic deformations: Extensions and reductions

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Abstract

We discuss the interpretation of dispersionless integrable hierarchies as equations of coisotropic deformations for certain associative algebras and other algebraic structures. We show that with this approach, the dispersionless Hirota equations for the dKP hierarchy are just the associativity conditions in a certain parameterization. We consider several generalizations and demonstrate that B-type dispersionless integrable hierarchies, such as the dBKP and the dVN hierarchies, are coisotropic deformations of the Jordan triple systems. We show that stationary reductions of the dispersionless integrable equations are connected with dynamical systems on the plane that are completely integrable on a fixed energy level.

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References

  1. V. E. Zakharov, Funct. Anal. Appl., 14, 89–98 (1980); Y. Kodama, Phys. Lett. A, 129, 223–226 (1988); B. A. Dubrovin and S. P. Novikov, Russ. Math. Surveys, 44, 35–124 (1989); B. A. Kuperschmidt, J. Phys. A, 23, 871–886 (1990); R. Carroll and Y. Kodama, J. Phys. A, 28, 6373–6387 (1995); J. Gibbons and S. P. Tsarev, Phys. Lett. A, 211, 19–24 (1996); P. B. Wiegmann and A. Zabrodin, Comm. Math. Phys., 213, 523–538 (2000).

    Article  MATH  Google Scholar 

  2. I. M. Krichever, Funct. Anal. Appl., 22, 200–213 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  3. Y. Kodama, Phys. Lett. A, 147, 477–482 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  4. K. Takasaki and T. Takebe, Internat. J. Mod. Phys. A, 7, 889–922 (1992).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. K. Takasaki and T. Takebe, Rev. Math. Phys., 7, 743–808 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  6. I. M. Krichever, Comm. Pure Appl. Math., 47, 437–475 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  7. V. E. Zakharov, “Dispersionless limit of integrable systems in 2+1 dimensions,” in: Singular Limits of Dispersive Waves (NATO Adv. Sci. Inst. Ser. B, Phys., Vol. 320, N. M. Ercolani et al., eds.), Plenum, New York (1994), pp. 165–174.

    Google Scholar 

  8. A. Boyasky, A. Marshakov, O. Ruchayskiy, P. B. Wiegmann, and A. Zabrodin, Phys. Lett., 515, 483–492 (2001).

    MathSciNet  Google Scholar 

  9. B. G. Konopelchenko and L. Martinez Alonso, J. Math. Phys., 43, 3807–3823 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  10. B. G. Konopelchenko and L. Martínez Alonso, Stud. Appl. Math., 109, 313–336 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  11. B. G. Konopelchenko and F. Magri, “Coisotropic deformations of associative algebras and dispersionless integrable hierarchies,” arXiv:nlin/0606069v2 [nlin.SI] (2006).

  12. A. Weinstein, J. Math. Soc. Japan, 40, 705–727 (1988); R. Berndt, An Introduction to Symplectic Geometry (Grad. Stud. Math., Vol. 26), Amer. Math. Soc., Providence, R. I. (2001).

    Article  MATH  MathSciNet  Google Scholar 

  13. G. Falqui, F. Magri, and M. Pedroni, Comm. Math. Phys., 197, 303–324 (1998).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. E. Witten, Nucl. Phys. B, 340, 281–332 (1990); R. Dijkgraaf, H. Verlinde, and E. Verlinde, Nucl. Phys. B, 352, 59–86 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  15. L. C. Li, Comm. Math. Phys., 203, 573–592 (1999).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. M. Błaszak, Phys. Lett. A, 297, 191–195 (2002).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. N. Jacobson, Structure and Representation of Jordan Algebras (Colloq. Publ., Vol. 39), Amer. Math. Soc., Providence, R. I. (1968); E. Neher, Jordan Triple Systems by the Grid Approach (Lect. Notes Math., Vol. 1280), Springer, Berlin (1987).

    Google Scholar 

  18. K. Takasaki, Lett. Math. Phys., 28, 177–185 (1993).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. V. V. Kozlov, Math. Notes, 45, 296–300 (1989).

    MATH  MathSciNet  Google Scholar 

  20. G. D. Birkhoff, Dynamical Systems, Amer. Math. Soc., New York (1927); E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies with an Introduction to the Problem of Three Bodies, Cambridge Univ. Press., Cambridge (1937).

    MATH  Google Scholar 

  21. J. Hietarinta, Phys. Rep., 147, 87–154 (1987).

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Fisica, Università degli studi di Lecce, INFN, Sezione di Lecce, 73100, Lecce, Italy

    B. G. Konopelchenko & F. Magri

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  1. B. G. Konopelchenko
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  2. F. Magri
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 439–457, June, 2007.

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Konopelchenko, B.G., Magri, F. Dispersionless integrable equations as coisotropic deformations: Extensions and reductions. Theor Math Phys 151, 803–819 (2007). https://doi.org/10.1007/s11232-007-0066-4

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