Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 079, 20 pages      arXiv:1902.08111      https://doi.org/10.3842/SIGMA.2019.079

Dispersionless Multi-Dimensional Integrable Systems and Related Conformal Structure Generating Equations of Mathematical Physics

Oksana Ye. Hentosh a, Yarema A. Prykarpatsky bc, Denis Blackmore d and Anatolij K. Prykarpatski e
a) Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine, Lviv, 79060, Ukraine
b) Department of Applied Mathematics, University of Agriculture in Krakow, 30059, Poland
c) Institute of Mathematics of NAS of Ukraine, Kyiv, 01024, Ukraine
d) Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102 USA
e) Department of Physics, Mathematics and Computer Science, Cracow University of Technology, Cracow, 31155, Poland

Received April 08, 2019, in final form October 07, 2019; Published online October 14, 2019

Abstract
Using diffeomorphism group vector fields on $\mathbb{C}$-multiplied tori and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems that describe conformal structure generating equations of mathematical physics. An interesting modification of the devised Lie-algebraic approach subject to spatial-dimensional invariance and meromorphicity of the related differential-geometric structures is described and applied in proving complete integrability of some conformal structure generating equations. As examples, we analyze the Einstein-Weyl metric equation, the modified Einstein-Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations and the inverse first Shabat reduction heavenly equation. We also analyze the modified Plebański heavenly equations, the Husain heavenly equation and the general Monge equation along with their multi-dimensional generalizations. In addition, we construct superconformal analogs of the Whitham heavenly equation.

Key words: Lax-Sato equations; multi-dimensional integrable heavenly equations; Lax integrability; Hamiltonian system; torus diffeomorphisms; loop Lie algebra; Lie-algebraic scheme; Casimir invariants; $R$-structure; Lie-Poisson structure; conformal structures; multi-dimensional heavenly equations.

pdf (423 kb)   tex (23 kb)  

References

  1. Adams M.R., Harnad J., Previato E., Isospectral Hamiltonian flows in finite and infinite dimensions. I. Generalized Moser systems and moment maps into loop algebras, Comm. Math. Phys. 117 (1988), 451-500.
  2. Adler V.E., Shabat A.B., Model equation of the theory of solitons, Theoret. and Math. Phys. 153 (2007), 1373-1387, arXiv:0706.0075.
  3. Blackmore D., Hentosh O.Ye., Prykarpatski A.K., The novel Lie-algebraic approach to studying integrable heavenly type multi-dimensional dynamical systems, J. Gen. Lie Theory Appl. 11 (2017), 1000287, 19 pages.
  4. Blackmore D., Prykarpatsky A.K., Samoylenko V.Hr., Nonlinear dynamical systems of mathematical physics. Spectral and symplectic integrability analysis, World Sci. Publ. Co. Pte. Ltd., Hackensack, NJ, 2011.
  5. Błaszak M., Classical $R$-matrices on Poisson algebras and related dispersionless systems, Phys. Lett. A 297 (2002), 191-195.
  6. Bogdanov L.V., Dryuma V.S., Manakov S.V., Dunajski generalization of the second heavenly equation: dressing method and the hierarchy, J. Phys. A: Math. Theor. 40 (2007), 14383-14393, arXiv:0707.1675.
  7. Doubrov B., Ferapontov E.V., On the integrability of symplectic Monge-Ampère equations, J. Geom. Phys. 60 (2010), 1604-1616, arXiv:0910.3407.
  8. Doubrov B., Ferapontov E.V., Kruglikov B., Novikov V.S., On a class of integrable systems of Monge-Ampère type, J. Math. Phys. 58 (2017), 063508, 12 pages, arXiv:1701.02270.
  9. Dunajski M., Anti-self-dual four-manifolds with a parallel real spinor, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002), 1205-1222, arXiv:math.DG/0102225.
  10. Dunajski M., Mason L.J., Tod P., Einstein-Weyl geometry, the dKP equation and twistor theory, J. Geom. Phys. 37 (2001), 63-93, arXiv:math.DG/0004031.
  11. Faddeev L.D., Takhtajan L.A., Hamiltonian methods in the theory of solitons, Classics in Mathematics, Springer, Berlin, 2007.
  12. Ferapontov E.V., Kruglikov B.S., Dispersionless integrable systems in 3D and Einstein-Weyl geometry, J. Differential Geom. 97 (2014), 215-254, arXiv:1208.2728.
  13. Harnad J., Winternitz P., Classical and quantum integrable systems in $\widetilde{\mathfrak{gl}}(2)^{+*}$ and separation of variables, Comm. Math. Phys. 172 (1995), 263-285, arXiv:hep-th/9312035.
  14. Harnad J., Wisse M.A., Moment maps to loop algebras, classical $R$-matrices and integrable systems, in Quantum Groups Integrable Models and Statistical Systems (Kingston, Canada, July 13-17, 1992), Editors J. LeTourneux, L. Vinet, World Scientific, Singapore, 1993, 105-117, arXiv:hep-th/9301104.
  15. Harnad J., Wisse M.A., Loop algebra moment maps and Hamiltonian models for the Painlevé transcendants, in Mechanics Day (Waterloo, ON, 1992), Fields Inst. Commun., Vol. 7, Amer. Math. Soc., Providence, RI, 1996, 155-169, arXiv:hep-th/9305027.
  16. Hentosh O.Ye., Prykarpatsky Ya.A., Lax-Sato integrable heavenly equations on functional supermanifolds and their Lie-algebraic structure, Eur. J. Math., to appear.
  17. Hentosh O.Ye., Prykarpatsky Ya.A., Blackmore D., Prykarpatski A.K., Lie-algebraic structure of Lax-Sato integrable heavenly equations and the Lagrange-d'Alembert principle, J. Geom. Phys. 120 (2017), 208-227.
  18. Mañas M., Medina E., Martínez Alonso L., On the Whitham hierarchy: dressing scheme, string equations and additional symmetries, J. Phys. A: Math. Gen. 39 (2006), 2349-2381, arXiv:nlin.SI/0509017.
  19. Manakov S.V., Santini P.M., On the solutions of the second heavenly and Pavlov equations, J. Phys. A: Math. Theor. 42 (2009), 404013, 11 pages, arXiv:0812.3323.
  20. Martínes Alonso L., Shabat A.B., Hydrodynamic reductions and solutions of a universal hierarchy, Theoret. and Math. Phys. 140 (2004), 1073-1085, arXiv:nlin.SI/0312043.
  21. Morozov O.I., A two-component generalization of the integrable rdDym equation, SIGMA 8 (2012), 051, 5 pages, arXiv:1205.1149.
  22. Ovsienko V., Bi-Hamiltonian nature of the equation $u_{tx}=u_{xy}u_y-u_{yy}u_x$, Adv. Pure Appl. Math. 1 (2010), 7-17, arXiv:0802.1818.
  23. Ovsienko V., Roger C., Looped cotangent Virasoro algebra and non-linear integrable systems in dimension $2+1$, Comm. Math. Phys. 273 (2007), 357-378, arXiv:math-ph/0602043.
  24. Plebański J.F., Some solutions of complex Einstein equations, J. Math. Phys. 16 (1975), 2395-2402.
  25. Pressley A., Segal G., Loop groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986.
  26. Reyman A.G., Semenov-Tian-Shansky M.A., Integrable systems, Computer Research Institute, Moscow - Izhevsk, 2003.
  27. Sergyeyev A., Szablikowski B.M., Central extensions of cotangent universal hierarchy: $(2+1)$-dimensional bi-Hamiltonian systems, Phys. Lett. A 372 (2008), 7016-7023, arXiv:0807.1294.
  28. Sheftel M.B., Yazıcı D., Bi-Hamiltonian representation, symmetries and integrals of mixed heavenly and Husain systems, J. Nonlinear Math. Phys. 17 (2010), 453-484, arXiv:0904.3981.
  29. Sheftel M.B., Yazıcı D., Recursion operators and tri-Hamiltonian structure of the first heavenly equation of Plebański, SIGMA 12 (2016), 091, 17 pages, arXiv:1712.01549.
  30. Szablikowski B.M., Hierarchies of Manakov-Santini type by means of Rota-Baxter and other identities, SIGMA 12 (2016), 022, 14 pages, arXiv:1512.05817.

Previous article  Next article  Contents of Volume 15 (2019)