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Conservation Laws and Potential Symmetries for a Generalized Gardner Equation

  • Rafael de la RosaEmail author
  • Tamara M. Garrido
  • María Santos Bruzón
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 18)

Abstract

In this paper, a generalized Gardner equation with nonlinear terms of any order has been analyzed from the point of view of group transformations and conservation laws. The generalized Gardner equation appears in many areas of physics and it is widely used to model a great variety of wave phenomena in plasma and solid state. By using the direct method of the multipliers, we have obtained an exhaustive classification of all low-order conservation laws which the generalized Gardner equation admits. Then, taking into account these conserved vectors we have determined the associated potential systems and we have searched for potential symmetries of the equation. Furthermore, we have determined and examined its first-level and second-level potential systems. From the first-level potential system we have found two new nonlocal conserved vectors.

Keywords

Partial differential equations Conservation laws Symmetries Potential symmetries 

Notes

Acknowledgements

The authors gratefully acknowledge the financial support from the Universidad Politécnica de Cartagena. The authors also acknowledge the financial support from Junta de Andalucía group FQM-201 and they express their sincere gratitude to the Plan Propio de Investigación de la Universidad de Cádiz.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Rafael de la Rosa
    • 1
    Email author
  • Tamara M. Garrido
    • 1
  • María Santos Bruzón
    • 1
  1. 1.Departamento de MatemáticasUniversidad de CádizCádizSpain

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