Abstract
This paper presents a general result on approximate conservation laws of perturbed partial differential equations. A method of constructing approximate conservation laws to systems of perturbed partial differential equations is given, which is based on approximate Noether symmetries of approximate and standard adjoint systems of the original system. The relationship between the Noether symmetry operators of approximate and standard adjoint system is established. As a result, the approach is applied to the perturbed wave equation and the perturbed KdV equation.
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References
Laplace, P.S.: Celestrial Mechanics, p. 1798. New York (1966)
Noether, E.: Invariante Variationsprobleme. Nacr. Konig. Gesell. Wissen., Gottingen, Math.-Phys. KI. Heft 2, 235–275 (1918) (English translation in Transp. Theory Stat. Phys. 1, 186–207 (1971))
Steudel, H.: Uber die Zuordnung zwischen Invarianzeigenschaften und Erhaltungssatzen. Z. Naturforsch. A 17, 129–132 (1962)
Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1993)
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)
Anco, S.C., Bluman, G.W.: Direct construction method for conservation laws of partial differential equations, part II: general treatment. Eur. J. Appl. Math. 9, 567–585 (2002)
Kara, A.H., Mahomed, F.M.: Noether-type symmetries and conservation laws via partial Lagrangians. Nonlinear Dyn. 45, 367–383 (2006)
Ibragimov, N.H.: A new conservation theorem. J. Math. Anal. Appl. 333, 311–328 (2007)
Kara, A.H., Mahomed, F.M., Ünal, G.: Approximate symmetries and conservation laws with applications. Int. J. Theor. Phys. 38, 2389–2999 (1999)
Johnpillai, A.G., Kara, A.H., Mahomed, F.M.: A basis of approximate conservation laws for PDEs with a small parameter. Int. J. Non-Linear Mech. 41, 830–837 (2006)
Johnpillai, A.G., Kara, A.H., Mahomed, F.M.: Approximate Noether-type symmetries and conservation laws via partial Lagrangians for PEDs with a small parameter. J. Comput. Appl. Math. 223, 508–518 (2009)
Ibragimov, N.H., Kara, A.H., Mahomed, F.M.: Lie–Bäcklund and Noether symmetries with applications. Nonlinear Dyn. 15, 115–136 (1998)
Johnpillai, A.G., Kara, A.H.: Variational formulation of approximate symmetries and conservation laws. Int. J. Theor. Phys. 40, 1501–1509 (2001)
Baikov, V.A., Ibragimov, N.H.: In: Ibragimov, N.H. (ed.) CRC Handbook of Lie Group of Analysis of Differential Equations, vol. 3. CRC Press, Boca Raton (1996)
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Gan, Y., Qu, C. Approximate conservation laws of perturbed partial differential equations. Nonlinear Dyn 61, 217–228 (2010). https://doi.org/10.1007/s11071-009-9643-4
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DOI: https://doi.org/10.1007/s11071-009-9643-4