Symmetries and conservation laws of evolution equations via multiplier and nonlocal conservation methods

Abstract

In this work, we have applied a new technique which is a union of multiplier and Ibragimov’s nonlocal conservation method for constructing the local conservation laws of nonlinear evolution equations. One can conclude that the higher order solutions of adjoint equation can be obtained by the multiplier functions. The Lax equation and generalized Hirota-Satsuma coupled KdV system are chosen to illustrate the effectiveness of the method. Thus, we have obtained a plenty of local (some of them are the higher order) conservation laws. The combined method presents a wider applicability for handling the conservation laws of nonlinear wave equations.

Keywords

• Lax equation,generalized Hirota-Satsuma coupled KdV system,symmetry,conservation laws

References

1. G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations, vol. 154 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2002.
2. P. J. Olver, Application of Lie groups to Differential Equations, Springer-Verlag, New York, 1993.
3. N. H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, Volume 1, Symmetries, Exact Solutions and Conservation Laws., CRC Press, Boca Raton, Florida, 1995.
4. A. F. Cheviakov, Gem software package for computation of symmetries and conservation laws of differential equations. Comput Phys Commun 2007; 176:48-61.
5. E. Noether, Invariante Variationsprobleme, Nacr. Konig. Gesell. Wissen., Gottingen, Math.-Phys. Kl. Heft 2 (1918) 235–257 (English translation in Transport Theory and Statistical Physiscs 1 (3) (1971) 186–207)
6. R. Naz,F. M. Mahomed, D. P. Mason, Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics, Applied Mathematics and Computation, 205 (2008) 212–230.
7. H. Steudel, Uber die zuordnung zwischen invarianzeigenschaften und erhaltungssatzen, Z. Naturforsch. 17A (1962) 129–132.
8. A. H. Kara, F. M. Mahomed, Relationship between symmetries and conservation laws, Int. J. Theor. Phys. 39 (2000) 23–40.

9. A. H. Kara, F. M. Mahomed, Noether-type symmetries and conservation laws via partial Lagragians, Nonlinear Dynam., 45 (2006) 367-383.
10. S. C. Anco, G W Bluman, Direct construction method for conservation laws of partial differential equations. Part I: examples of conservation law classifications, Eur. J. Appl. Math., 13 (2002) 545-566.
11. N.H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl. 333 (2007) 311–328.
12. T. Wolf, A comparison of four approaches to the calculation of conservation laws, Eur. J. Appl. Math. 13 (2002) 129–152.
13. N. H. Ibragimov, Nonlinear self-adjointness and conservation laws, Journal of Physics A: Mathematical and Theoretical 44.43 (2011) 432002.
14. W. Gangwei ,A. H. Kara, Conservation laws, multipliers, adjoint equations and Lagrangians for Jaulent–Miodek and some families of systems of KdV type equations, Nonlinear Dyn 81 (2015) 753–763.
15. P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Selected Papers Volume I (2005): 366-389.
16. A. M. Wazwaz, The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, Applied Mathematics and Computation 184.2 (2007): 1002-1014.
17. U. Goktas, W. Hereman, Symbolic computation of conserved densities for systems of nonlinear evolution equations, J. Symbol. Comput. 24 (1997) 591–621.
18. Y.T. Wu, X.G. Geng, X.B. Hu, S.M. Zhu, A generalized Hirota–Satsuma coupled Korteweg–de Vries equation and Miura transformations Phys. Lett. A 255 (1999) 259.
19. E. Fan, Soliton solutions for a generalized Hirota–Satsuma coupled KdV equation and a coupled MKdV equation, Physics Letters A 282 (2001) 18–22
20. A. H. Bokhari,A. Y. Al-Dweik, A. H. Kara,F. M. Mahomed,F. D. Zaman, Double reduction of a nonlinear (2+1) wave equation via conservation laws, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011) 1244–1253.