The Pfeiffer-Lax-Sato
type vector field equations and the related integrable versal deformations

D. Blackmore; A. Prykarpatski; M. Vovk; P. Pukach, Ya. Prykarpatsky; Ya. Prykarpatsky

doi:10.15330/ms.48.1.124-133

We study versal deformatiions of the Pfeiffer-Lax-Sato type vector field equations, related with a centrally extended metrized Lie algebra as the direct sum of vector fields and differential forms on torus.

1. R. Abraham, J. Marsden, Foundations of Mechanics, Second Edition, NY, Benjamin Cummings, 1978.

2. V.I. Arnold, Mathematical methods of classical mechanics, Springer, 1989.

3. V.I. Arnold, On matrices depending on parameters, Russian Math. Surveys, 26 (1971), 29–43.

4. D. Blackmore, A.K. Prykarpatsky, Versal deformations of a Dirac type differential operator, Journal of Nonlinear Mathematical Physics, 6 (1999), №3, 246–254.

5. D. Blackmore, A.K. Prykarpatsky, V.H. Samoylenko, Nonlinear dynamical systems of mathematical physics, World Scientific Publisher, NJ, USA, 2011.

6. L.V. Bogdanov, V.S. Dryuma, S.V. Manakov, Dunajski generalization of the second heavenly equation: dressing method and the hierarchy, J. Phys. A: Math. Theor., 40 (2007), 14383–14393.

7. L.V. Bogdanov, M.V. Pavlov, Linearly degenerate hierarchies of quasiclassical SDYM type, arXiv:1603.00238v2 [nlin.SI], 2016.

8. M.A. Buhl, Surles operateurs differentieles permutables ou non, Bull. des Sc. Math., 1928, S.2, t. LII, 353–361.

9. M.A. Buhl, Apercus modernes sur la theorie des groupes continue et finis, Mem. des Sc. Math., fasc. XXXIII, Paris, 1928.

10. M.A. Buhl, Apercus modernes sur la theorie des groupes continue et finis, Mem. des Sc. Math., fasc. XXXIII, Paris, 1928.

11. A. Cartan, Differential Forms, Dover Publisher, USA, 1971.

12. Earl A. Coddington, Norman Levinson, Theory of ordinary differential equations, International series in pure and applied mathematics, McGraw-Hill, 1955.

13. C. Godbillon, Geometrie Differentielle et Mecanique Analytique, Hermann Publ., Paris, 1969.

14. A. Edelman, E. Elmrothy, B. Kagstrom, A Geometric Approach to Perturbation Theory of Matrices and Matrix Pencils, Part I: Versal Deformations, March 23, 1995, Preprint: UMINF-95.09.

15. O.E. Hentosh, Ya.A. Prykarpatsky, D. Blackmore, A.K. Prykarpatski, Lie-algebraic structure of Lax– Sato integrable heavenly equations and the Lagrange–d’Alembert principle, Journal of Geometry and Physics, 120 (2017), 208–227.

16. B.G. Konopelchenko, Grassmanians $Gr(N-1,N+1),$ closed differential $N-1$ forms and $N$-dimensional integrable systems, arXiv:1208.6129v2 [nlin.SI], 2013.

17. F. Magri, Acta Applicandea Mathematica, 41 (1995), 247–270.

18. V. Ovsienko, Bi-Hamilton nature of the equation $u_{tx}=u_{xy}u_{y}-u_{yy}u_{x},$; arXiv:0802.1818v1, 2008.

19. V. Ovsienko, C. Roger, Looped cotangent virasoro algebra and non-linear integrable systems in dimension 2 + 1, Commun. Math. Phys., 273 (2007), 357–378.

20. G. Pfeiffer, Generalisation de la methode de Jacobi pour l’integration des systems complets des equations lineaires et homogenes, Comptes Rendues de l’Academie des Sciences de l’URSS, 190 (1930), 405–409.

21. M.G. Pfeiffer, Sur la operateurs d’un systeme complet d’equations lineaires et homogenes aux derivees partielles du premier ordre d’une fonction inconnue, Comptes Rendues de l’Academie des Sciences de l’URSS, 190 (1930), 909–911.

22. M.G. Pfeiffer, La generalization de methode de Jacobi-Mayer, Comptes Rendues de l’Academie des Sciences de l’URSS, 191 (1930), 1107–1109.

23. M.G. Pfeiffer, Sur la permutation des solutions s’une equation lineaire aux derivees partielles du premier ordre, Bull. des Sc. Math., LII (1928), S.2, 353–361.

24. M.G. Pfeiffer, Quelgues additions au probleme de M. Buhl, Atti dei Congresso Internationale dei Matematici, Bologna, III (1928), 45–46.

25. M.G. Pfeiffer, La construction des operateurs d’une equation lineaire, homogene aux derivees partielles premier ordre, Journal du Cycle Mathematique, Academie des Sciences d’Ukraine, Kyiv, 1 (1931), 37–72. (in Ukrainian)

26. C. Popovici, Sur les fonctions adjointes de M. Buhl, Comptes Rendus, 145 (1907), 749.

27. A. Prykarpatsky, D. Blackmore, Versal deformations of a Dirac type differential operator, Journal of Nonlinear Mathematical Physics, 6 (1999), №3, 246–254.

28. A.G. Reyman, M.A. Semenov-Tian-Shansky, Integrable systems, The Computer Research Institute Publ., Moscow-Izhvek, 2003. (in Russian)

29. K. Takasaki, T. Takebe, SDiff(2) Toda equation – Hierarchy, Tau function, and symmetries, Letters in Mathematical Physics, 23 (1991) №2, 205–214.

30. K. Takasaki, T. Takebe, Integrable Hierarchies and dispersionless limit, Reviews in Mathematical Physics, 7 (1995), №05, 743–808.

31. L.A. Takhtadjian, L.D. Faddeev, Hamiltonian approach in soliton theory, Springer, Berlin-Heidelberg, 1987.

	The Pfeiffer-Lax-Sato type vector field equations and the related integrable versal deformations
Author	D. Blackmore, A. Prykarpatski, M. Vovk, P. Pukach, Ya. Prykarpatsky denblac@gmail.com; pryk.anat@cybergal.com; mira.vovk1@gmail.com; pryk.anat@cybergal.com 1) Department of Mathematical Sciences at NJIT University Heights, Newark, NJ, USA; 2)The Department of Physics, Mathematics and Computer Science of the Cracov University of Technology, Krakow, Poland; 3, 4) Fundamental Sciences Institute Lviv Polytechnical Universtity, Lviv, Ukraine; 5) The Department of Applied Mathematics the University of Agriculture in Krakow, Poland
Abstract	We study versal deformatiions of the Pfeiffer-Lax-Sato type vector field equations, related with a centrally extended metrized Lie algebra as the direct sum of vector fields and differential forms on torus.
Keywords	torus diffeomorphisms; vector fields; differential forms; versal deformation; central extension; Lie- Poisson structures; Casimir invariants; central extension
DOI	doi:10.15330/ms.48.2.124-133
Reference	1. R. Abraham, J. Marsden, Foundations of Mechanics, Second Edition, NY, Benjamin Cummings, 1978. 2. V.I. Arnold, Mathematical methods of classical mechanics, Springer, 1989. 3. V.I. Arnold, On matrices depending on parameters, Russian Math. Surveys, 26 (1971), 29–43. 4. D. Blackmore, A.K. Prykarpatsky, Versal deformations of a Dirac type differential operator, Journal of Nonlinear Mathematical Physics, 6 (1999), №3, 246–254. 5. D. Blackmore, A.K. Prykarpatsky, V.H. Samoylenko, Nonlinear dynamical systems of mathematical physics, World Scientific Publisher, NJ, USA, 2011. 6. L.V. Bogdanov, V.S. Dryuma, S.V. Manakov, Dunajski generalization of the second heavenly equation: dressing method and the hierarchy, J. Phys. A: Math. Theor., 40 (2007), 14383–14393. 7. L.V. Bogdanov, M.V. Pavlov, Linearly degenerate hierarchies of quasiclassical SDYM type, arXiv:1603.00238v2 [nlin.SI], 2016. 8. M.A. Buhl, Surles operateurs differentieles permutables ou non, Bull. des Sc. Math., 1928, S.2, t. LII, 353–361. 9. M.A. Buhl, Apercus modernes sur la theorie des groupes continue et finis, Mem. des Sc. Math., fasc. XXXIII, Paris, 1928. 10. M.A. Buhl, Apercus modernes sur la theorie des groupes continue et finis, Mem. des Sc. Math., fasc. XXXIII, Paris, 1928. 11. A. Cartan, Differential Forms, Dover Publisher, USA, 1971. 12. Earl A. Coddington, Norman Levinson, Theory of ordinary differential equations, International series in pure and applied mathematics, McGraw-Hill, 1955. 13. C. Godbillon, Geometrie Differentielle et Mecanique Analytique, Hermann Publ., Paris, 1969. 14. A. Edelman, E. Elmrothy, B. Kagstrom, A Geometric Approach to Perturbation Theory of Matrices and Matrix Pencils, Part I: Versal Deformations, March 23, 1995, Preprint: UMINF-95.09. 15. O.E. Hentosh, Ya.A. Prykarpatsky, D. Blackmore, A.K. Prykarpatski, Lie-algebraic structure of Lax– Sato integrable heavenly equations and the Lagrange–d’Alembert principle, Journal of Geometry and Physics, 120 (2017), 208–227. 16. B.G. Konopelchenko, Grassmanians $Gr(N-1,N+1),$ closed differential $N-1$ forms and $N$-dimensional integrable systems, arXiv:1208.6129v2 [nlin.SI], 2013. 17. F. Magri, Acta Applicandea Mathematica, 41 (1995), 247–270. 18. V. Ovsienko, Bi-Hamilton nature of the equation $u_{tx}=u_{xy}u_{y}-u_{yy}u_{x},$; arXiv:0802.1818v1, 2008. 19. V. Ovsienko, C. Roger, Looped cotangent virasoro algebra and non-linear integrable systems in dimension 2 + 1, Commun. Math. Phys., 273 (2007), 357–378. 20. G. Pfeiffer, Generalisation de la methode de Jacobi pour l’integration des systems complets des equations lineaires et homogenes, Comptes Rendues de l’Academie des Sciences de l’URSS, 190 (1930), 405–409. 21. M.G. Pfeiffer, Sur la operateurs d’un systeme complet d’equations lineaires et homogenes aux derivees partielles du premier ordre d’une fonction inconnue, Comptes Rendues de l’Academie des Sciences de l’URSS, 190 (1930), 909–911. 22. M.G. Pfeiffer, La generalization de methode de Jacobi-Mayer, Comptes Rendues de l’Academie des Sciences de l’URSS, 191 (1930), 1107–1109. 23. M.G. Pfeiffer, Sur la permutation des solutions s’une equation lineaire aux derivees partielles du premier ordre, Bull. des Sc. Math., LII (1928), S.2, 353–361. 24. M.G. Pfeiffer, Quelgues additions au probleme de M. Buhl, Atti dei Congresso Internationale dei Matematici, Bologna, III (1928), 45–46. 25. M.G. Pfeiffer, La construction des operateurs d’une equation lineaire, homogene aux derivees partielles premier ordre, Journal du Cycle Mathematique, Academie des Sciences d’Ukraine, Kyiv, 1 (1931), 37–72. (in Ukrainian) 26. C. Popovici, Sur les fonctions adjointes de M. Buhl, Comptes Rendus, 145 (1907), 749. 27. A. Prykarpatsky, D. Blackmore, Versal deformations of a Dirac type differential operator, Journal of Nonlinear Mathematical Physics, 6 (1999), №3, 246–254. 28. A.G. Reyman, M.A. Semenov-Tian-Shansky, Integrable systems, The Computer Research Institute Publ., Moscow-Izhvek, 2003. (in Russian) 29. K. Takasaki, T. Takebe, SDiff(2) Toda equation – Hierarchy, Tau function, and symmetries, Letters in Mathematical Physics, 23 (1991) №2, 205–214. 30. K. Takasaki, T. Takebe, Integrable Hierarchies and dispersionless limit, Reviews in Mathematical Physics, 7 (1995), №05, 743–808. 31. L.A. Takhtadjian, L.D. Faddeev, Hamiltonian approach in soliton theory, Springer, Berlin-Heidelberg, 1987.
Pages	124-133
Volume	48
Issue	2
Year	2017
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

The Pfeiffer-Lax-Sato type vector field equations and the related integrable versal deformations