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Construction of Partial Differential Equations with Conditional Symmetries

  • Decio LeviEmail author
  • Miguel A. Rodríguez
  • Zora Thomova
Chapter
Part of the CRM Series in Mathematical Physics book series (CRM)

Abstract

Nonlinear PDEs having given conditional symmetries are constructed. They are obtained starting from the invariants of the conditional symmetry generator and imposing the extra condition given by the characteristic of the symmetry. Series of examples of new equations, constructed starting from the conditional symmetries of Boussinesq, are presented and discussed thoroughly to show and clarify the methodology introduced.

Keywords

Lie symmetries Partial differential equations Conditional symmetries Point transformations Boussinesq equation 

Notes

Acknowledgements

DL has been supported by INFN IS-CSN4 Mathematical Methods of Nonlinear Physics. DL thanks ZT and the SUNY Polytechnic Institute for their warm hospitality at Utica when this work was started. DL thanks the Departamento de Física Téorica of the Complutense University in Madrid for its hospitality. MAR was supported by the Spanish MINECO under project FIS 2015-63966-P. D. Nedza, summer student of ZT, contributed to the verification of some of the computations.

References

  1. 1.
    M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972)zbMATHGoogle Scholar
  2. 2.
    D.J. Arrigo, B.P. Ashley, S.J. Bloomberg, T.W. Deatherage, Nonclassical symmetries of a nonlinear diffusion–convection/wave equation and equivalents systems. Symmetry 8, 140 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    N. Bîlă, J. Niesen, On a new procedure for finding nonclassical symmetries. J. Symb. Comput. 38, 1523–1533 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    G.W. Bluman, Use and construction of potential symmetries. Math. Comput. Model. 18, 1–14 (1993)MathSciNetCrossRefGoogle Scholar
  5. 5.
    G.W. Bluman, J.D. Cole, The general similarity solutions of the heat equation. J. Math. Mech. 18, 1025–1042 (1969)MathSciNetzbMATHGoogle Scholar
  6. 6.
    G.W. Bluman, S. Kumei, Symmetries of Differential Equations (Springer, New York, 2002)zbMATHGoogle Scholar
  7. 7.
    G.W. Bluman, S. Kumei, G.J. Reid, New classes of symmetries for partial differential equations. J. Math. Phys. 29, 806–811 (1988)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    J. Boussinesq, Théorie de l’intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire. Comptes Rendus 72, 755–759 (1871)zbMATHGoogle Scholar
  9. 9.
    J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 7, 55–108 (1872)MathSciNetzbMATHGoogle Scholar
  10. 10.
    D. Catalano Ferraioli, Nonlocal aspects of λ-symmetries and ODEs reduction. J. Phys. A: Math. Theor. 40, 5479–5489 (2007)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    T. Chaolu, G. Bluman, An algorithmic method for showing existence of nontrivial nonclassical symmetries of partial differential equations without solving determining equations. J. Math. Anal. Appl. 411, 281–296 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    P.A. Clarkson, Nonclassical symmetry reductions of the Boussinesq equation. Chaos Solitons Fractals 5, 2261–2301 (1995)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    P.A. Clarkson, M.D. Kruskal, New similarity reductions of the Boussinesq equation. J. Math. Phys. 30, 2201–2213 (1989)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    P.A. Clarkson, E.L. Mansfield, Symmetry reductions and exact solutions of shallow water wave equations. Acta Appl. Math. 39, 245–276 (1995)MathSciNetCrossRefGoogle Scholar
  15. 15.
    W.I. Fushchich, Conditional symmetry of the equations of nonlinear mathematical physics. Ukr. Math. J. 43, 1350–1364 (1991)MathSciNetCrossRefGoogle Scholar
  16. 16.
    W.I. Fushchich, R.Z. Zhdanov, Conditional symmetry and reduction of partial differential equations. Ukr. Math. J. 44, 875–886 (1993)MathSciNetCrossRefGoogle Scholar
  17. 17.
    R.K. Gupta, M. Singh, Nonclassical symmetries and similarity solutions of variable coefficient coupled KdV system using compatibility method. Nonlinear Dyn. 87, 1543–1552 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    M.S. Hashemi, M.C. Nucci, Nonclassical symmetries for a class of reaction-diffusion equations: the method of heir-equations J. Nonlinear Math. Phys. 20, 44–60 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    R. Hernández–Heredero, E.G. Reyes, Nonlocal symmetries and a Darboux transformation for the Camassa-Holm equation. J. Phys. A: Math. Theor. 42, 182002 (2009)Google Scholar
  20. 20.
    L. Ji, C.Z. Qu, S. Shen, Conditional Lie-Backlund symmetry of evolution system and application for reaction-diffusion system. Stud. Appl. Math. 133, 118–149 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    I.S. Krasil’shchik, A.M. Vinogradov, Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations. Symmetries of partial differential equations, Part I. Acta Appl. Math. 15, 161–209 (1989)Google Scholar
  22. 22.
    D. Levi, P. Winternitz, Nonclassical symmetry reduction: example of the Boussinesq equation. J. Phys. A: Math. Gen. 22, 2915–2924 (1989)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    D. Levi, M.C. Nucci, M.A. Rodríguez, λ symmetries for the reduction of continuous and discrete equations. Acta Appl. Math. 122, 311–321 (2012)Google Scholar
  24. 24.
    C. Muriel, J.L. Romero, New methods of reduction for ordinary differential equations. IMA J. Appl. Math. 66, 111–125 (2001)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    M.C. Nucci, P.A. Clarkson, The nonclassical method is more general than the direct method for symmetry reductions. An example of the Fitzhugh-Nagumo equation. Phys. Lett. A 164, 49–56 (1992)Google Scholar
  26. 26.
    P.J. Olver, Applications of Lie Groups to Differential Equations (Springer, New York, 1993)CrossRefGoogle Scholar
  27. 27.
    R.O. Popovych, N.M. Ivanova, O.O. Vaneeva, Potential nonclassical symmetries and solutions of fast diffusion equation. Phys. Lett. A 362, 166–173 (2007)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    G.J. Reid, A.D. Wittkopf, A. Boulton, Reduction of systems of nonlinear partial differential equations to simplified involutive forms. Eur. J. Appl. Math. 7, 604–635 (1996)MathSciNetCrossRefGoogle Scholar
  29. 29.
    E.G. Reyes, Nonlocal symmetries and the Kaup-Kupershmidt equation. J. Math. Phys. 46, 073507 (2005)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    E.G. Reyes, On nonlocal symmetries of some shallow water equations. J. Phys. A: Math. Theor. 40, 4467–4476 (2007)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    P.M.M. Rocha, F.C. Khannab, T.M. Rocha Filhoa, A.E. Santana, Non-classical symmetries and invariant solutions of non-linear Dirac equations. Commun. Nonlinear Sci. Num. Simul. 26, 201–210 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    A.C. Scott, in Bäcklund Transformations, ed. by R. M. Miura. Lecture Notes in Mathematics, vol. 515 (Springer, Berlin, 1975), pp. 80–105Google Scholar
  33. 33.
    A. Sergyeyev, Constructing conditionally integrable evolution systems in (1+1) dimensions: a generalization of invariant modules approach. J. Phys. A: Math. Gen. 35, 7653–7660 (2002)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    A. Sergyeyev, On the classification of conditionally integrable evolution systems in (1+1) dimensions. J. Math. Sci. 136, 4392–4400 (2006)MathSciNetCrossRefGoogle Scholar
  35. 35.
    M. Toda, Studies of a nonlinear lattice. Phys. Rep. 18, 1–125 (1975)ADSCrossRefGoogle Scholar
  36. 36.
    A.M. Vinogradov, I.S. Krasil’shchik, A method of calculating higher symmetries of nonlinear evolutionary equations, and nonlocal symmetries (Russian). Dokl. Akad. Nauk SSSR 253(6), 1289–1293 (1980)ADSMathSciNetGoogle Scholar
  37. 37.
    N.J. Zabusky, A synergetic approach to problems of nonlinear dispersive wave propagation and interaction, in Nonlinear Partial Differential Equations, ed. by W.F. Ames (Academic, New York, 1967), pp. 233–258Google Scholar
  38. 38.
    V.E. Zakharov, On stochastization of one-dimensional chains of nonlinear oscillations. Sov. Phys. JETP 38, 108–110 (1974)ADSGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Decio Levi
    • 1
    Email author
  • Miguel A. Rodríguez
    • 2
  • Zora Thomova
    • 3
  1. 1.INFN, Sezione Roma TreRomaItaly
  2. 2.Dept. de Física TeóricaUniversidad Complutense de MadridMadridSpain
  3. 3.SUNY Polytechnic InstituteUticaUSA

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