Naz, R.; Naeem, I.; Abelman, S. Conservation laws for Camassa-Holm equation, Dullin-Gottwald-Holm equation and generalized Dullin-Gottwald-Holm equation. (English) Zbl 1179.35275 Nonlinear Anal., Real World Appl. 10, No. 6, 3466-3471 (2009). Summary: We construct the conservation laws for the Camassa-Holm equation, the Dullin-Gottwald-Holm equation (DGH) and the generalized Dullin-Gottwald-Holm equation (generalized DGH). The variational derivative approach is used to derive the conservation laws. Only first order multipliers are considered. Two multipliers are obtained for the Camassa-Holm equation. For the DGH and generalized DGH equations the variational derivative approach yields two multipliers; thus two conserved vectors are obtained. Cited in 8 Documents MSC: 35Q51 Soliton equations 35C05 Solutions to PDEs in closed form 35A15 Variational methods applied to PDEs Keywords:variational derivative approach; Camassa-Holm equation; Dullin-Gottwald-Holm equation; generalized Dullin-Gottwald-Holm equation; multipliers PDFBibTeX XMLCite \textit{R. Naz} et al., Nonlinear Anal., Real World Appl. 10, No. 6, 3466--3471 (2009; Zbl 1179.35275) Full Text: DOI References: [1] Cammasa, R.; Holm, D. D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 8, 1661-1664 (1993) · Zbl 0972.35521 [2] Ju, L., On Solution of the Dullin-Gottwald-Holm equation, Int. J. Nonlinear Sci., 1, 1, 43-48 (2006) · Zbl 1394.35419 [3] Tian, L.; Fang, G.; Gui, G., Well-posedness and blowup for an integrable shallow water equation with strong dispersive term, Int. J. Nonlinear Sci., 1, 1, 3-13 (2006) · Zbl 1394.35439 [4] Yin, J., Painleve Integrability, Bäcklund transformation and solitary solutions stability of modified DGH equation, Int. J. Nonlinear Sci., 2, 3, 183-187 (2006) · Zbl 1394.35452 [5] Tian, L.; Song, X., New peaked solitary wave solutions of the generalized Camassa-Holm equation, Chaos Solitons Fractals, 19, 3, 621-639 (2004) · Zbl 1068.35123 [6] Ding, D.; Tian, L., The study of solution of dissipative Camassa-Holm equation on total space, Int. J. Nonlinear Sci., 1, 1, 37-42 (2006) · Zbl 1394.35409 [7] Dullin, H. R.; Gottwald, G. A.; Holm, D. D., An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett., 87, 19, 4501-4504 (2001) [8] Tian, L.; Shi, Q., Boundary control of viscous Dullin-Gottwald-Holm equation, Int. J. Nonlinear Sci., 4, 1, 67-75 (2007) · Zbl 1394.93109 [9] Lua, D.; Pengb, D.; Tian, L., On the well-posedness problem for the generalized Dullin-Gottwald-Holm equation, Int. J. Nonlinear Sci., 1, 3, 178-186 (2006) · Zbl 1394.35426 [10] Anco, S. C.; Bluman, G. W., Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications, European J. Appl. Math., 13, 545-566 (2002) · Zbl 1034.35070 [11] Steudel, H., Uber die Zuordnung zwischen invarianzeigenschaften und Erhaltungssatzen, Z. Naturforsch., 17A, 129-132 (1962) [12] Olver, P. J., Applications of Lie Groups to Differential Equations (1993), Springer: Springer New York · Zbl 0785.58003 [13] 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 [14] R. Naz, D.P. Mason, F.M. Mahomed, Conservation laws and conserved quantities for laminar two-dimensional and radial jets, Nonlinear Anal. RWA doi:10.1016/j.nonrwa.2008.07.003; R. Naz, D.P. Mason, F.M. Mahomed, Conservation laws and conserved quantities for laminar two-dimensional and radial jets, Nonlinear Anal. RWA doi:10.1016/j.nonrwa.2008.07.003 · Zbl 1177.35171 [15] I. Naeem, R. Naz, Wall jet on a Hemi-spherical shell: Conserved quantities and group invariant solution, Int. J. Nonlinear Sci., 2008 (in press); I. Naeem, R. Naz, Wall jet on a Hemi-spherical shell: Conserved quantities and group invariant solution, Int. J. Nonlinear Sci., 2008 (in press) · Zbl 1176.35139 [16] Wolf, T., A comparison of four approaches to the calculation of conservation laws, European J. Appl. Math., 13, 129-152 (2002) · Zbl 1002.35008 [17] 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. Symbolic Comput., 27, 221-238 (1999) · Zbl 0919.65076 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.