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Approximate conservation laws of perturbed partial differential equations

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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

  1. Laplace, P.S.: Celestrial Mechanics, p. 1798. New York (1966)

  2. 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))

    Google Scholar 

  3. Steudel, H.: Uber die Zuordnung zwischen Invarianzeigenschaften und Erhaltungssatzen. Z. Naturforsch. A 17, 129–132 (1962)

    MathSciNet  Google Scholar 

  4. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1993)

    MATH  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  6. 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)

    MathSciNet  Google Scholar 

  7. Kara, A.H., Mahomed, F.M.: Noether-type symmetries and conservation laws via partial Lagrangians. Nonlinear Dyn. 45, 367–383 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ibragimov, N.H.: A new conservation theorem. J. Math. Anal. Appl. 333, 311–328 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kara, A.H., Mahomed, F.M., Ünal, G.: Approximate symmetries and conservation laws with applications. Int. J. Theor. Phys. 38, 2389–2999 (1999)

    Article  MATH  Google Scholar 

  10. 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)

    Article  MATH  MathSciNet  Google Scholar 

  11. 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)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ibragimov, N.H., Kara, A.H., Mahomed, F.M.: Lie–Bäcklund and Noether symmetries with applications. Nonlinear Dyn. 15, 115–136 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. Johnpillai, A.G., Kara, A.H.: Variational formulation of approximate symmetries and conservation laws. Int. J. Theor. Phys. 40, 1501–1509 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. 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)

    Google Scholar 

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Authors and Affiliations

  1. Center for Nonlinear Studies, Northwest University, Xi’an, 710069, China

    Yani Gan & Changzheng Qu

  2. Department of Mathematics, Northwest University, Xi’an, 710069, China

    Yani Gan & Changzheng Qu

Authors
  1. Yani Gan
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  2. Changzheng Qu
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Correspondence to Changzheng Qu.

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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

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