Skip to main content
SpringerLink
Log in

New Completely Integrable Dispersionless Dynamical System of Heavenly Type Generated By Vector Fields on a Torus

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

We study the problem of construction of a new spatially two-dimensional and dispersionless Lax–Sato completely integrable heavenly type equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. F. Plebanski, “Some solutions of complex Einstein equation,” J. Math. Phys., 16, Issue 12, 2395–2402 (1975).

    Article  MathSciNet  Google Scholar 

  2. L. M. Alonso and and A. B. Shabat, “Hydrodynamic reductions and solutions of a universal hierarchy,” Theor. Math. Phys., 104, 1073–1085 (2004).

    Article  MathSciNet  Google Scholar 

  3. D. Blackmore, O. E. Hentosh, and A. K. Prykarpatski, “The novel Lie-algebraic approach to studying integrable heavenly type multi-dimensional dynamical systems,” J. Generaliz. Lie Theory Appl., 11, 1000287 (2017).

    Google Scholar 

  4. M. Błaszak and B. Szablikowski, “Classical R-matrix theory of dispersionless: II. (2 + 1)-dimension theory,” J. Phys. A, 35, 10345 (2002).

    Article  MathSciNet  Google Scholar 

  5. L. V. Bohdanov, V. S. Dryuma, and S. V. Manakov, “Dunajski generalization of the second heavenly equation: dressing method and the hierarchy,” J. Phys. A, 40, 14383–14393 (2007).

    Article  MathSciNet  Google Scholar 

  6. M. Dunajski andW. Kryński, Einstein–Weyl Geometry, Dispersionless Hirota Equation and VeroneseWebs, Preprint arXiv:1301.0621 (2013).

  7. O. E. Hentosh, Y. A. Prykarpatsky, D. Blackmore, and A. K. Prykarpatski, “Lie-algebraic structure of Lax–Sato integrable heavenly equations and the Lagrange–d’Alembert principle,” J. Geom. Phys., 120, 208–227 (2017).

  8. O. E. Hentosh, Y. A. Prykarpatsky, A. Balinsky, and A. K. Prykarpatski, “The dispersionless completely integrable heavenly type Hamiltonian flows and their differential-geometric structure,” Ann. Math. Phys., 2(1), 11–25 (2019); DOI: https://doi.org/10.17352/amp.000006.

    Article  Google Scholar 

  9. O. E. Hentosh, Ya. A. Prikarpatsky, D. Blackmore, and A. K. Prykarpatski, “Dispersionless multi-dimensional integrable systems and related conformal structure generating equations of mathematical physics,” SIGMA Symmetry Integrability Geom. Methods Appl., 079 (2019).

  10. S. V. Manakov and P. M. Santini, “On the solution of the second heavenly and Pavlov equation,” J. Phys. A, 42, 404013 (2009).

    Article  MathSciNet  Google Scholar 

  11. O. I. Morozov and A. Sergyeyev, “The four-dimensional Martinez–Alonso–Shabat equation: reductions, nonlocal symmetries, and a four-dimensional integrable generalization of the ABC equation,” J. Geom. Phys., 85, 40–45 (2014).

    Article  MathSciNet  Google Scholar 

  12. M. Pavlov, “Kupershmidt hydrodynamic chains and lattices,” Int. Math. Res. Not.,” Art. ID 46987 (2006).

  13. M. Pavlov, “Integrable hydrodynamic chains,” J. Math. Phys., 44, Issue 9, 4134–4156 (2003).

    Article  MathSciNet  Google Scholar 

  14. Ya. A. Prykarpatsky and A. K. Prykarpatski, The Integrable Heavenly Type Equations and Their Lie-Algebraic Structure, Preprint arXiv: 1612.07760 [nlin.SI] (2016).

    Google Scholar 

  15. Ya. A. Prykarpatskyy and A. M. Samoilenko, “Classical M. A. Buhl problem, its Pfeiffer–Sato solutions, and the classical Lagrange–D’Alembert principle for the integrable heavenly-type nonlinear equations,” Ukr. Mat. Zh., 69, No. 12, 1652–1689 (2017); Ukr. Math. J., 69, No. 12, 1924–1967 (2018).

  16. M. M. Prytula, O. E. Hentosh, and Ya. A. Prykarpatskyy, “Differential-geometric structure and the Lax–Sato integrability of a class of dispersionless heavenly-type equations,” Ukr. Mat. Zh., 70, No. 2, 293–297 (2018); Ukr. Math. J., 70, No. 2, 334–339 (2018).

  17. B. Szablikowski and M. Błaszak, “Meromorphic Lax representations of .(1 + 1)-dimensional multi-Hamiltonian dispersionless systems,” J. Math. Phys., 47, No. 9, 092701 (2006).

Download references

Author information

Authors and Affiliations

  1. I. Franko National University of Lviv, Universytets’ka Str., 1, Lviv, 79602, Ukraine

    M. M. Prytula

Authors
  1. M. M. Prytula
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. M. Prytula.

Additional information

Translated from Neliniini Kolyvannya, Vol. 24, No. 1, pp. 110–127, January–March, 2021.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Prytula, M.M. New Completely Integrable Dispersionless Dynamical System of Heavenly Type Generated By Vector Fields on a Torus. J Math Sci 265, 682–702 (2022). https://doi.org/10.1007/s10958-022-06077-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-022-06077-3

Search

Navigation