The dispersionless completely integrable heavenly type Hamiltonian flows and their differential-eometric structure
Article Sidebar
Main Article Content
Abstract
There are reviewed modern investigations devoted to studying nonlinear dispersiveless heavenly type integrable evolutions systems on functional spaces within the modern differential-geometric and algebraic tools. Main accent is done on the loop diffeomorphism group vector fields on the complexified torus and the related Lie-algebraic structures, generating dispersionless heavenly type integrable systems. As examples, we analyzed the Einstein–Weyl metric equation, the modified Einstein–Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations, the inverse first Shabat reduction heavenly equation, the first and modified Plebański heavenly equations, the Husain heavenly equation, the general Monge equation and the classical Korteweg-de Vries dispersive dynamical system. We also investigated geometric structures of a class of spatially one-dimensional completely integrable Chaplygin type hydrodynamic systems, which proved to be deeply connected with differential systems on the complexified torus and the related diffeomorphism group orbits on them.
Downloads
Article Details
Copyright (c) 2019 Hentosh OE, et al.
This work is licensed under a Creative Commons Attribution 4.0 International License.
Licensing and protecting the author rights is the central aim and core of the publishing business. Peertechz dedicates itself in making it easier for people to share and build upon the work of others while maintaining consistency with the rules of copyright. Peertechz licensing terms are formulated to facilitate reuse of the manuscripts published in journals to take maximum advantage of Open Access publication and for the purpose of disseminating knowledge.
We support 'libre' open access, which defines Open Access in true terms as free of charge online access along with usage rights. The usage rights are granted through the use of specific Creative Commons license.
Peertechz accomplice with- [CC BY 4.0]
Explanation
'CC' stands for Creative Commons license. 'BY' symbolizes that users have provided attribution to the creator that the published manuscripts can be used or shared. This license allows for redistribution, commercial and non-commercial, as long as it is passed along unchanged and in whole, with credit to the author.
Please take in notification that Creative Commons user licenses are non-revocable. We recommend authors to check if their funding body requires a specific license.
With this license, the authors are allowed that after publishing with Peertechz, they can share their research by posting a free draft copy of their article to any repository or website.
'CC BY' license observance:
|
License Name |
Permission to read and download |
Permission to display in a repository |
Permission to translate |
Commercial uses of manuscript |
|
CC BY 4.0 |
Yes |
Yes |
Yes |
Yes |
The authors please note that Creative Commons license is focused on making creative works available for discovery and reuse. Creative Commons licenses provide an alternative to standard copyrights, allowing authors to specify ways that their works can be used without having to grant permission for each individual request. Others who want to reserve all of their rights under copyright law should not use CC licenses.
Das A (1989) Integrable Models. World Scientific 30:356. Link: http://bit.ly/2MERdUY
Drazin PG, Johnson RS (1989) Solitons: An Introduction. 2nd edition, Cambridge University Press. Link: http://bit.ly/2ZvnFv4
Novikov SP, Manakov SV, Pitaevskii LP, Zakharov VE (1984) Theory of Solitons. The Inverse Scattering Method. Springer. Link: http://bit.ly/2Zov536
Takhtajan LA, Faddeev LD (1987) Hamiltonian Approach in Soliton Theory, Springer, Berlin-Heidelberg.
Nizhnik LP (1988) The inverse scattering problems for the hyperbolic equations and their applications to non-linear integrable equations. Report Math Phys 26:261-283. Link: http://bit.ly/34cfGXo
Nizhnik LP (1996) Inverse scattering problem for the wave equation and its application. Parameter Identification and Inverse Problems in Hydrology, Geology and Ecology. Kluwer Academy Publisher 233-238.
Nizhnik LP, Pochynaiko MD (1982) The integration of a spatially two-dimensional Schrödinger equation by the inverse problem method. Functional Analysis and Its Applications 16:66-69. Link: http://bit.ly/30DqL1o
Samoilenko AM (2019) Theory of Multidimensional Delsarte–Lions Transmutation Operators. I. Ukrainian Mathematical Journal 70:1913–1952. Link: http://bit.ly/30Fa3i2
Samoilenko AM (2019) Theory of Multidimensional Delsarte–Lions Transmutation Operators. II. Ukrainian Mathematical Journal 71:808-839. Link: http://bit.ly/327YD70
Plebański JF (1975) Some solutions of complex Einstein equations. J Math Phys 16:2395-2402. Link: http://bit.ly/2zku0Pl
Błaszak M (2002) Classical R-matrices on Poisson algebras and related dispersionless systems. Physics Letters A 297:191-195. Link: http://bit.ly/30F4MqW
Bogdanov LV, Dryuma VS, Manakov SV (2007) Dunajski generalization of the second heavenly equation: dressing method and the hierarchy. J Phys A Math Theor 40:14383-14393. Link: http://bit.ly/30DFOYX
Doubrov B, Ferapontov EV (2010) On the integrability of symplectic Monge-Ampère equations. J Geom Phys 60:1604-1616. Link: http://bit.ly/2ZkKqXr
Doubrov B, Ferapontov EV, Kruglikov B, Novikov VS (2017) On a class of integrable systems of Monge-Ampère type. J Math Phys. Link: http://bit.ly/2zsiZv6
Dunajski M (2002) Anti-self-dual four-manifolds with a parallel real spinor. Proc Roy Soc 458:1205-1222. Link: http://bit.ly/2NDtoMX
Dunajski M, Mason LJ, Tod P (2001) Einstein–Weyl geometry, the dKP equation and twistor theory. J Geom Phys 37:63-93. Link: http://bit.ly/2MGk4YV
Hentosh OE, Prykarpatsky YA, Blackmore D, Prykarpatski AK (2017) Lie-algebraic structure of Lax–Sato integrable heavenly equations and the Lagrange-d’Alembert principle. Journal of Geometry and Physics 120:208-227. Link: http://bit.ly/2ZhANsg
Manas M, Medina E, Martinez-Alonso L (2006) On the Whitham hierarchy: dressing scheme, string equations and additional symmetries. J Phys 39:2349-2381. Link: http://bit.ly/2MGjLgJ
Manakov SV, Santini PM (2009) On the solutions of the second heavenly and Pavlov equations. J Phys 42:404013. Link: http://bit.ly/2NDsXCj
Sergyeyev A, Szablikowski BM (2008) Central extensions of cotangent universal hierarchy: (2+1)-dimensional bi-Hamiltonian systems. Phys Lett 372:7016-7023. Link: http://bit.ly/2MDOM5a
Sheftel MB, Yazıcı D (2010) Bi-Hamiltonian representation, symmetries and integrals of mixed heavenly and Husain systems. Link: http://bit.ly/2ZsCouN
Brunelli JC, Gürses M, Zheltukhin K (1999) On the Integrability of a Class of Monge-Ampère Equations. 13. Link: http://bit.ly/2zowNXJ
Pressley A, Segal G (1986) Loop groups, Clarendon Press, London.
Reyman AG, Semenov-Tian-Shansky MA (2003) Integrable Systems. The Computer Research Institute Publ., Moscow-Izhvek.
Blackmore D, Prykarpatsky AK, Samoylenko VH (2011) Nonlinear dynamical systems of mathematical physics. World Scientific Publisher, NJ, USA 564. Link: http://bit.ly/2NFzOLL
Ovsienko V (2008) Bi-Hamilton nature of the equation = ,u u u u utx xy y yy x−. arXiv:0802.1818v1.
Ovsienko V, Roger C (2007) Looped Cotangent Virasoro Algebra and Non-Linear Integrable Systems in Dimension 2+1. Commun Math Phys 273:357-378. Link: http://bit.ly/2ZxZ9JT
Szablikowski B (2016) Hierarchies of Manakov–Santini Type by Means of Rota-Baxter and Other Identities. SIGMA 12:022-036. Link: http://bit.ly/344L8GG
Ferapontov E, Kruglikov BS (2014) Dispersionless integrable systems in 3D and Einstein-Weyl geometry. J Differential Geometry 97:215-254. Link: http://bit.ly/2NwJbx7
Hentosh OE, Prykarpatsky YAA (2019) The Lax-Sato integrable heavenly equations on functional supermanifolds and their Lie-algebraic structure. European Journal of Mathematics 15. Link: http://bit.ly/345MxNh
Morozov OI (2012) A two-Component generalization of the integrable rd-Dym equation. SIGMA 8:051-055. Link: http://bit.ly/2Zm15tB
Arik M, Neyzi F, Nutku Y, Olver PJ, Verosky J (1988) Multi-Hamiltonian Structure of the Born-Infeld Equation. J Math Phys 30:1338. Link: http://bit.ly/32ayIv9
Olver PJ, Nutku Y (1988) Hamiltonian Structures for Systems of Hyperbolic Conservation Laws. J Math Phys 29:1610-1619. Link: http://bit.ly/342hyC0
Stanyukovich KP (1960) Unsteady Motion of Continuous Media. Pergamon, New York. 137. Link: http://bit.ly/2Laye1p
Sheftel MB, Yazıcı D (2017) Evolutionary Hirota type (2+1)-dimensional equations: Lax pairs, recursion operators and bi-Hamiltonian structures. SIGMA 14:17-19. Link: http://bit.ly/345ulTZ