You are here

2+1 BOYUTTA YENİ İNTEGRE EDİLEBİLİR HAMİLTONIAN SİSTEMLER

A NEW 2+1-DIMENSIONAL HAMILTONIAN INTEGRABLE SYSTEM

Journal Name:

Publication Year:

Keywords (Original Language):

Author NameUniversity of AuthorFaculty of Author

AMS Codes:

Abstract (2. Language): 
It is shown that a new 2+1-dimensional second-order partial differential equation, when written as a first-order nonlinear evolutionary system, admits bi-Hamiltonian structure. Therefore, by Magri’s theorem it is a completely integrable system. For this system a Lagrangian is introduced and Dirac’s theory is applied in order to obtain first Hamiltonian structure. Then recursion operator is constructed and finally the second Hamiltonian structure for this system is obtained. Jacobi identity for the Hamiltonian structure is proved by using Olver’s method. Thus, it is an example of a completely integrable system in three dimensions.
Abstract (Original Language): 
Yeni 2+1 boyutlu ikinci mertebeden kismi türevli diferansiyel denklem, birinci mertebe lineer olmayan değişim sistemi olarak yazıldığında, bu yeni sistemin bi-Hamiltonian yapıya sahip olduğu gösterilmiştir. Böylece Magri teoremine göre tamamen integere edilebilir bir sistem elde edilmiştir. Bu sistem için Lagrangian elde edilmiş, ve birinci Hamiltonian yapıyı elde etmek için Dirac teori uygulanmıştır. Sistem için tekrarlama (recursion) operatorü kurulmuş ve son olarak ikinci Hamiltonian yapı elde edilmiştir. Hamiltonian yapılar için Jacobi özdeşliği Olver’in metodu kullanılarak ispatlanmıştır. Böylece yeni denklem üç boyutta tamamen integre edilebilir sitemlere bir örnek teşkil etmektedir.

REFERENCES

References: 

[1] Gardner C. S., Korteweg-de Vrise equation and generalizations. IV. The Korteweg-de
Vrise equation as Hamiltonian system, J. Math. Phys. 12, 1548-1551, 1971.
[2] Zakharov V. E., Fadeev L. D., Korteweg-de Vrise equation: a completely integrable
Hamiltonian system, Func. Anal. Appl 5, 208-287, 1971.
[3] Magri F., A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19,
1156-1162, 1978.
[4] Kupershmidt B. A., Geometry of jet bundles and structure of Lagrangian and
Hamiltonian formalism, in Geometric Methods in Mathematical Physics, G. Kaiser and J.
E. Marsden, eds., Lecture Notes in Math. No. 775, pp. 162-218, Springer-Verlag,
Newyork, 1980
[5] Manin Yu. I., Algebraic aspects of nonlinear differential equations, J. Soviet Math. 11, 1-
122, 1979.
[6] I. M. Gelfand and I. Ya. Dorfman, Hamiltonian operators and algebraic structures
related to them, Func. Anal. Appl. 13, 248-262, 1979.
[7] P. J. Olver, On the Hamiltonian structure of evaluation equations, Math. Proc. Camb. Phil.
Soc. 88, 71-88, 1980.
[8] Y. Kosmann –Schwarzbach, Hamiltonian system on fibered manifolds, Lett. Math. Phys.
5, 229-237, 1981.
[9] Magri F., A geometrical approach to the nonlinear solvable equations, in Nonlinear
Evolution Equations and Dynamic Systems, M. Boitti, F. Pempinelli and G. Soliani, eds.,
Lecture Notes in Physics, No. 120 Springer-Verlag, Newyork 1980.
[10] F. Neyzi, Y. Nutku and M. B. Sheftel, Multi Hamiltonian structure of Plebanski’s second
heavenly equation, J. Phys. A: Math. Gen. 38 8473-8485, 2005.
[11] Y. Nutku, M. B. Sheftel, J. Kalaycı and D. Yazıcı, Self-dual gravity is completely
integrable, J. Phys. A: Math. Gen. 41, 395206, 13pp, 2008.
[12] D. Yazıcı, M. B. Sheftel, Symmetry reductions of second heavenly equation and 2+1-
dimensional Hamiltonian integrable system, Journal of Nonlinear Mathematical Physics,
Volume 15,supplement 3, 417-425 (2008).
[13] Dirac P. A. M., Lecture Notes in Quantum Mechanics , Belfer Graduate School of
Science Monographs series 2, New York, 1964.
[14] P. J. Olver, Application of Lie groups to differential equations, Springer, New York,
1986.
[15] P. M. Santini and A. S. Fokas, Commun. Math. Physics., 115 , 375, 1988.
[16] Y. Nutku, Lagrangian approach to integrable systems yields new symplectic structure for
KdV, hep-th/0011052v1, 8 Nov 2008.

Thank you for copying data from http://www.arastirmax.com