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Conservation laws and hierarchies of potential symmetries for certain diffusion equations

Author

Listed:
  • Ivanova, N.M.
  • Popovych, R.O.
  • Sophocleous, C.
  • Vaneeva, O.O.

Abstract

We show that the so-called hidden potential symmetries considered in a recent paper [M.L. Gandarias, New potential symmetries for some evolution equations, Physica A 387 (2008) 2234–2242] are ordinary potential symmetries that can be obtained using the method introduced by Bluman and collaborators [G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, New York, 1989; G.W. Bluman, G.J. Reid, S. Kumei, New classes of symmetries for partial differential equations, J. Math. Phys. 29 (1988) 806–811]. In fact, these are simplest potential symmetries associated with potential systems which are constructed with single conservation laws having no constant characteristics. Furthermore we classify the conservation laws for classes of porous medium equations, and then using the corresponding conserved (potential) systems we search for potential symmetries. This is the approach one needs to adopt in order to determine the complete list of potential symmetries. The provenance of potential symmetries is explained for the porous medium equations by using potential equivalence transformations. Point and potential equivalence transformations are also applied to deriving new results on potential symmetries and corresponding invariant solutions from known ones. In particular, in this way the potential systems, potential conservation laws and potential symmetries of linearizable equations from the classes of differential equations under consideration are exhaustively described. Infinite series of infinite-dimensional algebras of potential symmetries are constructed for such equations.

Suggested Citation

  • Ivanova, N.M. & Popovych, R.O. & Sophocleous, C. & Vaneeva, O.O., 2009. "Conservation laws and hierarchies of potential symmetries for certain diffusion equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(4), pages 343-356.
    • Handle: RePEc:eee:phsmap:v:388:y:2009:i:4:p:343-356
    DOI: 10.1016/j.physa.2008.10.018
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    References listed on IDEAS

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    1. Gandarias, M.L., 2008. "New potential symmetries for some evolution equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(10), pages 2234-2242.
    2. Sophocleous, Christodoulos, 2005. "Further transformation properties of generalised inhomogeneous nonlinear diffusion equations with variable coefficients," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 345(3), pages 457-471.
    3. Sophocleous, Christodoulos, 2003. "Classification of potential symmetries of generalised inhomogeneous nonlinear diffusion equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 320(C), pages 169-183.
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    Cited by:

    1. Feng, Wei & Ji, Lina, 2013. "Conditional Lie–Bäcklund symmetries and functionally separable solutions of the generalized inhomogeneous nonlinear diffusion equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(4), pages 618-627.
    2. Tu, Jian-Min & Tian, Shou-Fu & Xu, Mei-Juan & Zhang, Tian-Tian, 2016. "On Lie symmetries, optimal systems and explicit solutions to the Kudryashov–Sinelshchikov equation," Applied Mathematics and Computation, Elsevier, vol. 275(C), pages 345-352.

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