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Conservation laws for a complexly coupled KdV system, coupled Burgers system and Drinfeld-Sokolov-Wilson system via multiplier approach. (English) Zbl 1221.37119

Summary: The multiplier approach (variational derivative method) is used to derive the conservation laws for some nonlinear systems of partial differential equations. Firstly, the multipliers (characteristics) are computed and then conserved vectors are obtained for the each multiplier. Examples of the third-order complexly coupled KdV system, second-order coupled Burgers’ system and third-order Drinfeld-Sokolov-Wilson system are considered. For all three systems the local conservation laws are established by utilizing the multiplier approach.

MSC:

37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
35A30 Geometric theory, characteristics, transformations in context of PDEs

Software:

GeM
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Full Text: DOI

References:

[1] Noether E. Invariante Variationsprobleme, Nachr. König. Gesell. Wiss. Göttingen. Math.-phys. Kl. Heft 2; 1918. p. 235-57 [English transl., Transport Theory Stat Phys 1971;1(3):186-207].; Noether E. Invariante Variationsprobleme, Nachr. König. Gesell. Wiss. Göttingen. Math.-phys. Kl. Heft 2; 1918. p. 235-57 [English transl., Transport Theory Stat Phys 1971;1(3):186-207].
[2] Laplace PS. Traité de Mécanique Céleste, vol. 1, Paris; 1978 [English transl., Celestial mechanics, New York; 1966].; Laplace PS. Traité de Mécanique Céleste, vol. 1, Paris; 1978 [English transl., Celestial mechanics, New York; 1966].
[3] Steudel, H., Uber die Zuordnung zwischen invarianzeigenschaften und Erhaltungssatzen, Z Naturforsch, 17A, 129-132 (1962)
[4] Olver, P. J., Applications of Lie groups to differential equations (1993), Springer: Springer New York, p. 435-58
[5] Anco, S. C.; Bluman, G. W., Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications, Eur J Appl Math, 13, 545-566 (2002) · Zbl 1034.35070
[6] Kara, A. H.; Mahomed, F. M., Relationship between symmetries and conservation laws, Int J Theor Phys, 39, 23-40 (2000) · Zbl 0962.35009
[7] Kara, A. H.; Mahomed, F. M., Noether-type symmetries and conservation laws via partial Lagrangians, Nonlinear Dynam, 45, 367-383 (2006) · Zbl 1121.70014
[8] Wolf, T., A comparison of four approaches to the calculation of conservation laws, Eur J Appl Math, 13, 129-152 (2002) · Zbl 1002.35008
[9] Wolf, T.; Brand, A.; Mohammadzadeh, M., Computer algebra algorithms and routines for the computation of conservation laws and fixing of gauge in differential expressions, J Symb Comput, 27, 221-238 (1999) · Zbl 0919.65076
[10] Göktas, Ü.; Hereman, W., Symbolic computation of conserved densities for systems of nonlinear evolution equations, J Symb Comput, 24, 591-621 (1997) · Zbl 0891.65129
[11] Hereman, W.; Adams, P. J.; Eklund, H. L.; Hickman, M. S.; Herbst, B. M., Direct methods and symbolic software for conservation laws of nonlinear equations, (Yan, Z., Advances of nonlinear waves and symbolic computation (2009), Nova Science: Nova Science New York), 19-79, [chapter 2] · Zbl 1210.35164
[12] Hereman, W.; Colagrosso, M.; Sayers, R.; Ringler, A.; Deconinck, B.; Nivala, M., Continuous and discrete homotopy operators and the computation of conservation laws, (Wang, D.; Zheng, Z., Differential equations with symbolic computation (2005), Birkhäuser: Birkhäuser Basel), 249-285
[13] Hereman, W., Symbolic computation of conservation laws of nonlinear partial differential equations in multi-dimensions, Int J Quant Chem, 106, 1, 278-299 (2006) · Zbl 1188.68364
[14] Cheviakov, A. F., GeM software package for computation of symmetries and conservation laws of differential equations, Comp Phys Commun, 176, 1, 48-61 (2007) · Zbl 1196.34045
[15] Naz, R.; Mahomed, F. M.; Mason, D. P., Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics, Appl Math comput, 205, 212-230 (2008) · Zbl 1153.76051
[16] Naz, R.; Mason, D. P.; Mahomed, F. M., Conservation laws and conserved quantities for laminar two-dimensional and radial jets, Nonlinear Anal, 10, 2641-2651 (2009) · Zbl 1177.35171
[17] Hirota, R.; Satsuma, J., Soliton solutions of a coupled Korteweg deVries equation, Phys Lett A, 85, 407-408 (1981)
[18] Guha, P., Geodesic flows, bi-Hamiltonian structure and coupled KdV type systems, J Math Anal Appl, 310, 45-56 (2005) · Zbl 1076.35110
[19] Inan, I. E.; Kaya, D., Exact solutions of some nonlinear partial differential equations, Physica A, 318, 104-115 (2007)
[20] Abdou, M. A.; Soliman, A. A., Variational iteration method for solving Burgers’ and coupled Burgers’ equations, J Comp Appl Math, 181, 2, 245-251 (2005) · Zbl 1072.65127
[21] Abassy, T. A.; El-Tawil, M. A.; El-Zoheiry, H., Exact solutions of some nonlinear partial differential equations using the variational iteration method linked with Laplace transforms and the Padé technique, Comp Math Appl, 54, 940-954 (2007) · Zbl 1141.65382
[22] Hirota, R.; Grammaticos, B.; Ramani, A., Soliton structure of the Drinfeld-Sokolov-Wilson equation, J Math Phys, 27, 1499-1505 (1986) · Zbl 0638.35071
[23] Bekir, A., Applications of the extended tanh method for coupled nonlinear evolution equations, Commun Non Sci Numer Simulat, 13, 1748-1757 (2008) · Zbl 1221.35322
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