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Geometric Structures on the Orbits of Loop Diffeomorphism Groups and Related Heavenly-Type Hamiltonian Systems. I

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Ukrainian Mathematical Journal Aims and scope

We present a review of differential-geometric and Lie-algebraic approaches to the investigation of a broad class of nonlinear integrable differential systems of “heavenly” type associated with Hamiltonian flows on the spaces conjugate to the loop Lie algebras of vector fields on the tori. These flows are generated by the corresponding orbits of the coadjoint action of loop diffeomorphism groups and satisfy the Lax–Satotype vector-field compatibility conditions. We analyze the corresponding hierarchies of conservation laws and their relationships with Casimir invariants. We consider typical examples of these systems and establish their complete integrability by using the developed Lie-algebraic construction. We also describe new generalizations of the integrable dispersion-free systems of heavenly type for which the corresponding generating elements of the orbits have factorized structures, which allows their extension to the multidimensional case.

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References

  1. M. Arik, F. Neyzi, Y. Nutku, P. J. Olver, and J. Verosky, “Multi-Hamiltonian structure of the Born–Infeld equation,” J. Math. Phys., 30 (1988).

  2. N. N. Bogoliubov, Yu. A. Mitropolski, and A. M. Samoilenko, Method of Accelerated Convergence in Nonlinear Mechanics [in Russian], Naukova Dumka, Kyiv (1969).

    Google Scholar 

  3. J. C. Brunelli, M. G¨urses, and K. Zheltukhin, On the Integrability of a Class of Monge–Ampère Equations, Preprint arXiv:hepth/9906233v1 29 Jun 1999.

  4. A. Das, Integrable Models, World Scientific, Singapore (1989).

    Book  MATH  Google Scholar 

  5. B. Doubrov and E. V. Ferapontov, “On the integrability of symplectic Monge–Amp`ere equations,” J. Geom. Phys., 60, No. 10, 1604–1616 (2010); Preprint arXiv:0910.3407v2 [math.DG] 13 Apr 2010.

  6. B. Doubrov, E. V. Ferapontov, B. Kruglikov, and V. S. Novikov, On a Class of Integrable Systems of Monge–Amp`ere Type, Preprint arXiv:1701.02270v1 [nlin.SI] 9 Jan 2017.

  7. P. G. Drazin and R. S. Johnson, Solitons: an Introduction, Cambridge Univ. Press, Cambridge (1989).

    Book  MATH  Google Scholar 

  8. E. Ferapontov and B. S. Kruglikov, “Dispersionless integrable systems in 3D and Einstein–Weyl geometry,” J. Different. Geom., 97, 215–254 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  9. E. V. Ferapontov and J. Moss, Linearly Degenerate PDEs and Quadratic Line Complexes, Preprint arXiv:1204.2777v1 [math.DG] 12 Apr 2012.

  10. O. E. Hentosh and Ya. A. Prykarpatsky, “The Lax–Sato integrable heavenly equations on functional supermanifolds and their Liealgebraic structure,” Europ. J. Math. (2018).

  11. M. Manas, E. Medina, and L. Martinez-Alonso, “On the Whitham hierarchy: dressing scheme, string equations, and additional symmetries,” J. Phys. A: Math. Gen., 39, 2349–2381 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  12. O. I. Morozov, “A two-component generalization of the integrable rd-Dym equation,” SIGMA, 8, Article 051 (2012).

  13. L. P. Nizhnik, “The inverse scattering problems for the hyperbolic equations and their applications to non-linear integrable equations,” Rep. Math. Phys., 26, No. 2, 261–283 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  14. L. P. Nizhnik, “Inverse scattering problem for the wave equations and its applications,” in: Parameter Identification and Inverse Problems in Hydrology, Geology and Ecology, Kluwer AP, Dordrecht (1996), pp. 233–238.

  15. L. P. Nizhnik and M. D. Pochynaiko, “The integration of a spatially two-dimensional Schrödinger equation by the inverse problem method,” Func. Anal. Appl., 16, No. 1, 80–82 (1982).

    Article  Google Scholar 

  16. S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons. The Inverse Scattering Method, Springer (1984).

    MATH  Google Scholar 

  17. P. J. Olver and Y. Nutku, “Hamiltonian structures for systems of hyperbolic conservation laws,” J. Math. Phys., 29 (1988).

  18. A. M. Samoilenko, Ya. A. Prykarpatsky, D. Blackmore, and A. K. Prykarpatsky, “Theory of multidimensional Delsarte–Lions transmutation operators. I,” Ukr. Mat. Zh., 70, No. 12, 1660–1695 (2018); English translation: Ukr. Math. J., 70, No. 12, 1913–1952 (2019).

  19. A. M. Samoilenko, Ya. A. Prykarpatsky, D. Blackmore, and A. K. Prykarpatsky, “Theory of multidimensional Delsarte–Lions transmutation operators. II,” Ukr. Mat. Zh., 71, No. 6, 808–839 (2019); English translation: Ukr. Math. J., 71, No. 6, 921–955 (2019).

  20. B. Szablikowski, “Hierarchies of Manakov–Santini type by means of Rota–Baxter and other identities,” SIGMA, 12, Article 022 (2016).

  21. M. B. Sheftel and D. Yazıcı, Bi-Hamiltonian Representation, Symmetries and Integrals of Mixed Heavenly and Husain Systems, Preprint arXiv:0904.3981v4 [math-ph] 4 May 2010.

  22. M. B. Sheftel and D. Yazıcı, “Evolutionary Hirota type (2 + 1)-dimensional equations: Lax pairs, recursion operators and bi-Hamiltonian structures,” SIGMA, 14, Article 017 (2018); Preprint arXiv:1712.01549v1 [math-ph] 5 Dec 2017.

  23. L. A. Takhtajan and L. D. Faddeev, Hamiltonian Approach in Soliton Theory, Springer, Berlin (1987).

    Google Scholar 

  24. P. P. Kulish, “Analog of the Korteweg–de Vries equation for a superconformal algebra,” Zap. Nauch. Sem. LOMI, 155, 142–148 (1986).

    Google Scholar 

  25. V. G. Mikhalev, “On the Hamiltonian formalism of the Korteweg–de Vries-type hierarchies,” Funkts. Anal. Prilozh., 26, No. 2, 79–82 (1992).

    MathSciNet  MATH  Google Scholar 

  26. V. Ovsienko and C. Roger, “Looped cotangent Virasoro algebra and non-linear integrable systems in dimension 2+1,” Comm. Math. Phys., 273, No. 2, 357–378 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  27. V. Ovsienko, “Bi-Hamiltonian nature of the equation utx = uxyuy −uyyux,” Adv. Pure Appl. Math., 1, No. 1, 7–10 (2008); Preprint arXiv:0802.1818v1 (2008).

  28. A. Sergyeyev and B. M. Szablikowski, “Central extensions of cotangent universal hierarchy: (2 + 1)-dimensional bi-Hamiltonian systems,” Phys. Lett. A, 372, No. 47, 7016–7023 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  29. A. K. Prykarpatski, O. Ye. Hentosh, and Ya. A. Prykarpatsky, “The differential-geometric and algebraic aspects of the Lax–Sato theory,” Mathematics, 5, No. 4, MDPI, Basel, Switzerland (2017).

  30. O. Ye. Hentosh, Ya. A. Prykarpatsky, D. Blackmore, and A. K. Prykarpatski, “Dispersionless completely integrable heavenly type Hamiltonian flows and their differential-geometric structure,” Symmetry, Integrability, Geom.: Meth. Appl., 15, Article 079 (2019); https://doi.org/10.3842/SIGMA.2019.079.

  31. O. Ye. Hentosh, Ya. A. Prykarpatsky, D. Blackmore, and A. K. Prykarpatski, “Lie-algebraic structure of Lax–Sato integrable heavenly equations and the Lagrange–d’Alembert principle,” 120, Article 208 (2017); https://doi.org/10.1016/j.geomphys.2017.06.003.

  32. J. F. Plebański, “Some solutions of complex Einstein equations,” J. Math. Phys., 16, No. 12, 2395–2402 (1975).

    Article  MathSciNet  Google Scholar 

  33. S. V. Manakov and P. M. Santini, “Inverse scattering problem for vector fields and the Cauchy problem for the heavenly equation,” Phys. Lett. A, 359, No. 6, 613–619 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  34. L. V. Bogdanov, V. S. Dryuma, and S. V. Manakov, “Dunajski generalization of the second heavenly equation: dressing method and the hierarchy,” J. Phys. A: Math. Theor., 40, No. 48, 14383–14393 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  35. M. Dunajski, “Anti-self-dual four-manifolds with a parallel real spinor,” Proc. Roy. Soc. A, 458, 1205–1222 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  36. M. Dunajski, L. J. Mason, and P. Tod, “Einstein–Weyl geometry, the dKP equation and twistor theory,” J. Geom. Phys., 37, No. 1-2, 63–93 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  37. M. P. Pavlov, “Integrable hydrodynamic chains,” J. Math. Phys., 44, No. 9, 4134–4156 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  38. W. K. Schief, “Self-dual Einstein spaces via a permutability theorem for the Tzitzeica equation,” Phys. Lett. A, 223, No. 1-2, 55–62 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  39. W. K. Schief, “Self-dual Einstein spaces and a discrete Tzitzeica equation. A permutability theorem link,” in: Symmetr. Integr. Difference Equat., London Math. Soc., Lect. Notes Ser., 255 (1999), pp. 137–148.

  40. K. Takasaki and T. Takebe, “SDi↵(2) Toda equation—hierarchy, tau function, and symmetries,” Lett. Math. Phys., 23, No. 3, 205–214 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  41. K. Takasaki and T. Takebe, “Integrable hierarchies and dispersionless limit,” Rev. Math. Phys., 7, No. 5, 743–808 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  42. A. G. Reiman and M. A. Semenov-Tyan-Shanskii, Integrable Systems. Group-Theoretic Approach [in Russian], Institute of Computer Investigations, Izhevsk (2003).

  43. M. Blaszak, “Classical R-matrices on Poisson algebras and related dispersionless systems,” Phys. Lett. A, 297, No. 3-4, 191–195 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  44. M. Blaszak and B. M. Szablikowski, “Classical R-matrix theory of dispersionless systems: II. (2+1) dimension theory,” J. Phys. A: Math. Gen., 35, No. 48, 10345–10364 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  45. D. Blackmore, A. K. Prykarpatsky, and V. Hr. Samoylenko, Nonlinear Dynamical Systems of Mathematical Physics: Spectral and Symplectic Integrability Analysis, World Scientific, Hackensack (2011).

    Book  MATH  Google Scholar 

  46. A. Alekseev and A. Z. Malkin, “Symplectic structure of the moduli space of flat connection on a Riemann surface,” Comm. Math. Phys., 169, No. 1, 99–119 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  47. A. Pressley and G. Segal, Loop Groups, Clarendon Press, London (1988).

    MATH  Google Scholar 

  48. M. Audin, “Lectures on gauge theory and integrable systems,” in: Gauge Theory and Symplectic Geometry, Kluwer (1997), pp. 1–48.

  49. O. Hentosh and Ya. Prykarpatsky, “The Lax–Sato integrable heavenly equations on functional supermanifolds and their Lie-algebraic structure,” Europ. J. Math. (2019); https://doi.org/https://doi.org/10.1007/s40879-019-00329-4.

    Article  MATH  Google Scholar 

  50. I. A. B. Strachan and B. M. Szablikowski, “Novikov algebras and a classification of multicomponent Camassa–Holm equations,” Stud. Appl. Math., 133, 84–117 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  51. O. D. Artemovych, A. A. Balinsky, D. Blackmore, and A. K. Prykarpatski, “Reduced pre-Lie algebraic structures, the weak and weakly deformed Balinsky–Novikov type symmetry algebras and related Hamiltonian operators,” Symmetry, 10, Article 601 (2018).

  52. O. D. Artemovych, D. Blackmore, and A. K. Prykarpatski, “Examples of Lie and Balinsky–Novikov algebras related to Hamiltonian operators,” Topol. Algebra Appl., 6, No. 1, 43–52 (2018).

    MathSciNet  MATH  Google Scholar 

  53. O. D. Artemovych, D. Blackmore, and A. K. Prykarpatski, “Poisson brackets, Novikov–Leibniz structures, and integrable Riemann hydrodynamic systems,” J. Nonlin. Math. Phys., 24, No. 1, 41–72 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  54. D. Blackmore, Y. Prykarpatsky, J. Golenia, and A. Prykarpatski, “Hidden symmetries of Lax integrable nonlinear systems,” Appl. Math., 4, 95–116 (2013).

    Article  Google Scholar 

  55. M. A. Semenov-Tyan-Shanskii, “What is a classical r-matrix?,” Funkts. Anal. Prilozhen., 17, No. 4, 17–33 (1983).

    MathSciNet  Google Scholar 

  56. R. Abraham and J. Marsden, Foundations of Mechanics, Addison-Wesley Publ. Co., Redwood City, CA (1978).

    MATH  Google Scholar 

  57. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York (1978).

    Book  Google Scholar 

  58. V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Springer, New York (1998).

    Book  MATH  Google Scholar 

  59. T. Kambe, “Geometric theory of fluid flows and dynamical systems,” Fluid Dyn. Res., 30, 331–378 (2002).

    Article  MATH  Google Scholar 

  60. J. Marsden, T. Ratiu, and A.Weinstein, “Reduction and Hamiltonian structures on duals of semidirect product Lie algebras,” Contemp. Math., 28, 55–100 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  61. D. Holm, J. Marsden, T. Ratiu, and A.Weinstein, “Nonlinear stability of fluid and plasma equilibria,” Phys. Rep., 123, No. 1-2, 1–116 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  62. V. I. Arnold, “Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications `a l’hydrodynamique des fluides parfaits,” Ann. Inst. Fourier (Grenoble), 16, No. 1, 319–361 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  63. D. Holm and B. Kupershmidt, “Poisson structures of superfluids,” Phys. Lett. A, 91, 425–430 (1982).

    Article  MathSciNet  Google Scholar 

  64. E. A. Kuznetsov and A. V. Mikhailov, “On the topological meaning of canonical Clebsch variables,” Phys. Lett. A, 77, No. 1, 37–38 (1980).

    Article  MathSciNet  Google Scholar 

  65. B. A. Kupershmidt and T. Ratiu, “Canonical maps between semidirect products with applications to elasticity and superfluids,” Comm. Math. Phys., 90, 235–250 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  66. A. Weinstein, “Sophus Lie and symplectic geometry,” Expo. Math., 1, 95–96 (1983).

    MathSciNet  MATH  Google Scholar 

  67. A. Weinstein, “The local structure of Poisson manifolds,” J. Different. Geom., 18, No. 3, 523–557 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  68. J. Marsden and A. Weinstein, “Reduction of symplectic manifolds with symmetry,” Rep. Math. Phys., 5, No. 1, 121–130 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  69. L. V. Bogdanov, “Interpolating differential reductions of multidimensional integrable hierarchies,” Teor. Mat. Fiz., 167, No. 3, 705–713 (2011).

    Article  MathSciNet  Google Scholar 

  70. L. V. Bogdanov and B. G. Konopelchenko, “On the heavenly equation and its reductions,” J. Phys. A: Math. Gen., 39, 11793–11802 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  71. L. V. Bogdanov and M. V. Pavlov, “Linearly degenerate hierarchies of quasiclassical SDYM type,” J. Math. Phys., 58, No. 9 (2017).

  72. L. Martínez Alonso and A. B. Shabat, “Hydrodynamic reductions and the solutions of a universal hierarchy,” Teor. Mat. Fiz., 140, No. 2, 216–229 (2004).

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Authors and Affiliations

  1. Institute for Applied Problems in Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine

    O. E. Hentosh

  2. Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    Ya.A. Prykarpatskyy

  3. University of Agriculture, Kraków, Poland

    Ya.A. Prykarpatskyy

  4. Mathematics Institute, Cardiff University, Cardiff, UK

    A. A. Balinsky

  5. Institute of Mathematics, Kraków University of Technology, Kraków, Poland

    A. K. Prykarpatski

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  1. O. E. Hentosh
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  2. Ya.A. Prykarpatskyy
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  3. A. A. Balinsky
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  4. A. K. Prykarpatski
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Correspondence to Ya.A. Prykarpatskyy.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 8, pp. 1029–1059, August, 2022. Ukrainian https://doi.org/10.37863/umzh.v74i8.6614.

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Hentosh, O.E., Prykarpatskyy, Y., Balinsky, A.A. et al. Geometric Structures on the Orbits of Loop Diffeomorphism Groups and Related Heavenly-Type Hamiltonian Systems. I. Ukr Math J 74, 1175–1208 (2023). https://doi.org/10.1007/s11253-023-02129-2

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