ArticlesFree Access

BI-HAMILTONIAN REPRESENTATION, SYMMETRIES AND INTEGRALS OF MIXED HEAVENLY AND HUSAIN SYSTEMS

M. B. SHEFTEL

Physics Department, Boğaziçi University, 34342 Bebek, Istanbul, Turkey

Search for more papers by this author

and

D. YAZICI

Physics Department, Yıldız Technical University, 34220 Esenler, Istanbul, Turkey

Search for more papers by this author

https://doi.org/10.1142/S1402925110001021Cited by:10 (Source: Crossref)

Abstract

In the recent paper by one of the authors (MBS) and A. A. Malykh on the classification of second-order PDEs with four independent variables that possess partner symmetries [1], mixed heavenly equation and Husain equation appear as closely related canonical equations admitting partner symmetries. Here for the mixed heavenly equation and Husain equation, formulated in a two-component form, we present recursion operators, Lax pairs of Olver–Ibragimov–Shabat type and discover their Lagrangians, symplectic and bi-Hamiltonian structure. We obtain all point and second-order symmetries, integrals and bi-Hamiltonian representations of these systems and their symmetry flows together with infinite hierarchies of nonlocal higher symmetries.

Keywords:

References

M. B. Sheftel and A. A. Malykh, On classification of second-order PDEs possessing partner symmetries, J. Phys. A: Math. Theor. 42 (2009) 395202 (preprint) , arXiv:0904.2909v3 . Google Scholar
A. A. Malykh, Y. Nutku and M. B. Sheftel, J. Phys. A: Math. Gen. 36, 10023 (2003), DOI: 10.1088/0305-4470/36/39/304. Crossref, Google Scholar
A. A. Malykh, Y. Nutku and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004), DOI: 10.1088/0305-4470/37/30/010. Crossref, Google Scholar
A. A. Malykh, Y. Nutku and M. B. Sheftel, J. Phys. A: Math. Theor. 40, 9371 (2007), DOI: 10.1088/1751-8113/40/31/014. Crossref, Web of Science, Google Scholar
M. B. Sheftel and A. A. Malykh, J. Nonlinear Math. Phys. 15(), 375 (2008), DOI: 10.2991/jnmp.2008.15.s3.37. Google Scholar
P. J. Olver , Applications of Lie Groups to Differential Equations ( Springer , New York , 1986 ) . Crossref, Google Scholar
J. F. Plebañski, J. Math. Phys. 16, 2395 (1975). Crossref, Web of Science, Google Scholar
B. Doubrov and E. V. Ferapontov, On the integrability of symplectic Monge–Ampère equations, preprint (2010) , arXiv:0910.3407v2 . Google Scholar
V. Husain, Self-dual gravity as a two dimensional theory and conservation laws, Class. Quantum Grav. 11 (1994) 927–937 (preprint) , arXiv:gr-qc/9310003 . Google Scholar
J. F. Plebañski, M. Przanowski and H. García-Compeán, From proncipal chiral model to self-dual gravity, Mod. Phys. Lett. A 11 (1996) 663–674 (preprint) , arXiv:hep-th/9509092v2 . Google Scholar
P. J. Olver, J. Math. Phys. 18, 1212 (1977), DOI: 10.1063/1.523393. Crossref, Web of Science, Google Scholar
N. H. Ibragimov and A. B. Shabat, Funct. Anal. Appl. 14, 19 (1980). Crossref, Google Scholar
F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), 1156–1162; F. Magri, Nonlinear Evolution Equations and Dynamical Systems (Lecture Notes in Physics Vol. 120) ed. M. Boiti, F. Pempinelli and G. Soliani, (Springer, New York, 1980), p. 233 . Google Scholar
F. Neyzi, Y. Nutku and M. B. Sheftel, Multi-Hamiltonian structure of Plebanski's second heavenly equation, J. Phys. A: Math. Gen. 38 (2005) 8473–8485 (preprint) , arXiv:nlin.SI/0505030 . Google Scholar
Y. Nutku, M. B. Sheftel, J. Kalayci and D. Yazıcı, Self-dual gravity is completely integrable, J. Phys. A: Math. Theor. 41 (2008) 395206 (preprint) , arXiv:0802.2203 . Google Scholar
P. A. M. Dirac , Lectures on Quantum Mechanics , Belfer Graduate School of Science Monographs ( Yeshiva University Press , New York , 1964 ) . Google Scholar
T. Wolf, J. Comp. Phys. 60, 437 (1985), DOI: 10.1016/0021-9991(85)90029-4. Crossref, Web of Science, Google Scholar
N. H. Ibragimov , Transformation Groups Applied to Mathematical Physics ( Reidel , Boston , 1985 ) . Crossref, Google Scholar
B. Fuchssteiner and A. S. Fokas, Physica 4D, 47 (1981). Google Scholar
M. B. Sheftel, CRC handbook of Lie group analysis of differential equations, New Trends in Theoretical Developments and Computational Methods 3, ed. N. H. Ibragimov (CRC Press, Boca Raton, FL, 1996) pp. 91–137. Google Scholar
A. Ashtekar, Phys. Rev. D 36, 1587 (1987), DOI: 10.1103/PhysRevD.36.1587. Crossref, Web of Science, Google Scholar
A. Ashtekar, T. Jacobson and L. Smolin, Comm. Math. Phys. 115, 631 (1988), DOI: 10.1007/BF01224131. Crossref, Web of Science, Google Scholar
J. F. Plebañski and M. Przanowski, Phys. Lett. A 212, 22 (1996). Crossref, Web of Science, Google Scholar