Reduction of non-variational bi-Hamiltonian system of shallow-water waves propagation via symmetry approach

Yıl 2018, Cilt: 1 Sayı: 1, 39 - 44, 27.05.2018

Adil Jhangeer

https://doi.org/10.33187/jmsm.411423

Öz

In this paper, non-variational bi-Hamiltonian system of shallow-water waves propagation is considered. Lie point generators are calculated and one dimensional optimal system of its subalgebras up to conjugacy classes are reported. Then similarity variables are computed by using these conjugacy classes which are further utilized for the reduction of considered system. Then, a transformation is used to convert the system from non-variational to variational system, thus standard Lagrangian is computed. Noether operators are calculated by using Noether approach and local conserved quantity is discussed for the new fourth order system of partial differential equations (PDEs). Further, inverse transformation is applied to get the corresponding local conserved quantity for the considered non-variational problem. Moreover, this local conservation law with the help of double reduction theorem is utilized to reduce the system.

Anahtar Kelimeler

Lagrangian, Noether operator, Conserved quantities

Kaynakça

• [1] M. Ablowitz and H. Segur, Solitons and the inverse scattering transform, SIAM Philadelphia, 1981.
• [2] S. C. Anco and G. W. Blauman, Direct construction method for conservation laws of partial differential equation, Part II: General treatment, Euor. J. Appl. Math., 9 (2002);567-585:
• [3] A. H. Bokhari, A. Y. Al-Dweika, F. D. Zaman, A. H. Kara and F. M. Mahomed, Generalization of the double reduction theory, Nonlinear Anal.: Real World Appl., 11(5) (2010);3763-3769.
• [4] A. F. Cheviakov, GeM software package for computaion of symmetries and conservation laws of differential equation, Comp. Phys. Commun., 176 (2007);48-61.
• [5] Yu. A. Chirkunov, E. O. Pikmullina, Symmetry properties and solutions of shallow water equations, Universal J. Appl. Math., 2(1) (2014);10-23:
• [6] Ü. Göktaş, W. Hereman, Symbolic computation of conserved densties for system of nonlinear evolution equation, J. Symb. Comput., 24 (1997);591-621.
• [7] W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman, B. M. Herbst, Direct methods and symbolic software for conservation laws of nonlinear equations, In: Advances of Nonlinear Waves and Symbolic Computation, New york: Nova Science, Yan Z (Ed.), (2009);19-79.
• [8] W. Hereman, M. Colagrosso, R. Sayers, A. Ringler, B. Deconinck, M. Nivala et al., Continous and discrete homotopy operators and the computation of conservation laws, In: D. Wang, Z. Zheng (Ed.), Differential Equations with Symbolic Computation, Basel: Birkh ¨ auser, (2005);249-285.
• [9] W. Hereman, Symbolic computation of conservation laws of nonlinear partial differential equations in multi-dimensions, Int. J. Quant. Chem., 106 (2006);278-299.
• [10] A. H. Kara, F. M. Mahomed, Noether-type symmetries and conservation laws via partial Lagrangian, Nonlinear Dyn., 45 (2006);367-383.
• [11] F. Neyzi, M. B. Sheftel, D. Yazici, Symmetries, integrals and three dimensional reductions of Plebanski’s second heavenly equation, Phys. Atom. Nuclei, 70(3) (2007);584-592.
• [12] E. Noether, Invariante Variationsprobleme, Nacr. K¨onig. Gesell. Wissen., G¨ottingen, Math.-phys. Kl. Heft, 2 (1918); 235􀀀257, (English translation in Transport Theo. Stat. Phy., 1(3) (1971);186-207.
• [13] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986.
• [14] M. Pandey, Exact solutions of the shallow water equations, Intl. J. Nonlinear Sci., 16 (2013);334􀀀339.
• [15] A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equation, CRC Press, Boca Raton London, 2004.
• [16] J. Patera and P. Winternitz, Subalgebras of real three- and four-dimensional Lie algebras, J. Math. Phy., 18(7) (1977);1449􀀀1455:
• [17] T. R. Sekhar and B. Bira, Wave features and group analysis for axisymmetric flow of shallow water equations, Int. J. Nonlinear Sci., 14 (2012);23-30.
• [18] A. Sj¨oberg, Double reduction of PDEs from the association of symmetries with conservation laws with applications, Appl. Math. Comput., 84 (2007);608-616.
• [19] A. Sj¨oberg, On double reductions from symmetries and conservation laws, Non. Lin. Anal. Real World Appl., 10(6) (2009);3472􀀀3477.
• [20] H. Steudel, U¨ ber die Zuordnung zwischen invarianzeigenschaften und Erhaltungssatzen. Z Naturforsch, 17A (1962);29-32.
• [21] T. Wolf, A comparision of four approches to the calculation of conservation laws, Euorp. J. Appl. Math., 13 (2002);129-152.
• [22] T. Wolf, A. Brand, M. Mohammadzadeh, Computer algebra algorithems and routines for the computations of conservation laws and fixing of guage in differential expressions, J. Symb. Comput., 27 (1999);221-238.
• [23] D. Yazici, M. B. Sheftel, Symmetry reductions of second heavenly equations and 2+1 dimensional hamiltonian integrable system, J. Non. Lin. Math. Phys., 15(3) (2008);417-425.