Skip to main content
SpringerLink
Log in

Conservation laws, multipliers, adjoint equations and Lagrangians for Jaulent–Miodek and some families of systems of KdV type equations

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, we will show and integrate the ‘multiplier’ and ‘adjoint equations and Lagrangian’ approaches to the construction of conservation laws. The details will reveal how the method lends itself to higher-order solutions of the adjoint equation. Amongst other things, the differences here involves the determination of the solutions of the adjoint equation via the multipliers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ostrovsky, L.A., Grue, J.: Evolution equations for strongly nonlinear internal waves. Phys. Fluids 15(10), 2934–2948 (2003)

    Article  MathSciNet  Google Scholar 

  2. Jaulent, M., Miodek, I.: Nonlinear evolution equations associated with ‘energy dependent Shrödinger potentials’. Lett. Math. Phys. 1, 243–250 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  3. Guererro, F.G., Alonso, L.M.: Hamiltonian formulation for the Jaulent–Miodek family of nonlinear evolution equations. Lettere al Nuovo Cimento 27(1), 28–31 (1980)

    Article  MathSciNet  Google Scholar 

  4. Laddomada, C., Gui-Zhang, T.: Bäcklund transformations of the Jaulent–Miodek equations. Lett. Math. Phys. 6, 453–462 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ganji, D.D., Jannatabadi, M., Mohseni, E.: Application of Hes variational iteration method to nonlinear Jaulent–Miodek equations and comparing it with ADM. J. Comput. Appl. Math. 207, 35–45 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Kara, A.H.: A symmetry invariance analysis of the multipliers and conservation laws of the Jaulent–Miodek and some families of systems of KdV type equations. J. Nonlinear Math. Phys. 16, 149–156 (2009)

    Article  MathSciNet  Google Scholar 

  7. Liu, De-Yin, Tian, Bo, Jiang, Yan, Sun, Wen-Rong: Soliton solutions and Bäcklund transformations of a (2+1)-dimensional nonlinear evolution equation via the Jaulent–Miodek hierarchy. Nonlinear Dyn. 78, 2341–2347 (2014)

    Article  MathSciNet  Google Scholar 

  8. Krishnan, E.V., Triki, H., Labidi, M., Biswas, A.: A study of shallow water waves with Gardners equation. Nonlinear Dyn. 66, 497–507 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Smaoui, Nejib, Al-Jamal, Rasha H.: Boundary control of the generalized Kortewegde VriesBurgers equation. Nonlinear Dyn. 51, 439–446 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Noether, E.: Invariante variationsprobleme, Nachrichten der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse, 2 (1918) 235 (English translation in Transport Theory and Statistical Physics, 1 (1971) 186)

  11. Anco, S., Bluman, G.: Direct construction method for conservation laws of partial differential equations Part I: examples of conservation law classifications. Eur. J. Appl. Math. 13, 545 (2002)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Ibragimov, N.: Integrating factors, adjoint equations and Lagrangians. J. Math. Anal. Appl. 318, 742–757 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kara, A.H., Mahomed, F.M.: Relationship between symmetries and conservation laws Int. J. Theor. Phys. 39(1), 23–40 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kara, A.H., Mahomed, F.M.: A basis of conservation laws for partial differential equations. J. Nonlinear Math. Phys. 9, 60–72 (2002)

    Article  MathSciNet  Google Scholar 

  16. Euler, N., Euler, M.: On nonlocal symmetries and nonlocal conservation laws and nonlocal transformations of evolution equations: two linearisable hierarchies. J. Nonlinear Math. Phys. arXiv: 0811.1105

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, People’s Rebublic of China

    Gangwei Wang

  2. School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa

    A. H. Kara

  3. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

    A. H. Kara

Authors
  1. Gangwei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. H. Kara
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. H. Kara.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, G., Kara, A.H. Conservation laws, multipliers, adjoint equations and Lagrangians for Jaulent–Miodek and some families of systems of KdV type equations. Nonlinear Dyn 81, 753–763 (2015). https://doi.org/10.1007/s11071-015-2025-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-015-2025-1

Keywords

Search

Navigation