Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 106, 21 pages      arXiv:1212.6475      https://doi.org/10.3842/SIGMA.2012.106
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

Nonlocal Symmetries, Telescopic Vector Fields and λ-Symmetries of Ordinary Differential Equations

Concepción Muriel and Juan Luis Romero
Department of Mathematics, University of Cádiz, 11510 Puerto Real, Spain

Received July 09, 2012, in final form December 19, 2012; Published online December 28, 2012

Abstract
This paper studies relationships between the order reductions of ordinary differential equations derived by the existence of λ-symmetries, telescopic vector fields and some nonlocal symmetries obtained by embedding the equation in an auxiliary system. The results let us connect such nonlocal symmetries with approaches that had been previously introduced: the exponential vector fields and the λ-coverings method. The λ-symmetry approach let us characterize the nonlocal symmetries that are useful to reduce the order and provides an alternative method of computation that involves less unknowns. The notion of equivalent λ-symmetries is used to decide whether or not reductions associated to two nonlocal symmetries are strictly different.

Key words: nonlocal symmetries; λ-symmetries; telescopic vector fields; order reductions; differential invariants.

pdf (477 kb)   tex (30 kb)

References

  1. Abraham-Shrauner B., Hidden symmetries and nonlocal group generators for ordinary differential equations, IMA J. Appl. Math. 56 (1996), 235-252.
  2. Abraham-Shrauner B., Hidden symmetries, first integrals and reduction of order of nonlinear ordinary differential equations, J. Nonlinear Math. Phys. 9 (2002), suppl. 2, 1-9.
  3. Abraham-Shrauner B., Govinder K.S., Leach P.G.L., Integration of second order ordinary differential equations not possessing Lie point symmetries, Phys. Lett. A 203 (1995), 169-174.
  4. Abraham-Shrauner B., Guo A., Hidden and nonlocal symmetries of nonlinear differential equations, in Modern Group Analysis: Advanced Analytical and computational Methods in Mathematical Physics (Acireale, 1992), Kluwer Acad. Publ., Dordrecht, 1993, 1-5.
  5. Abraham-Shrauner B., Leach P.G.L., Hidden symmetries of nonlinear ordinary differential equations, in Exploiting Symmetry in Applied and Numerical Analysis (Fort Collins, CO, 1992), Lectures in Appl. Math., Vol. 29, Amer. Math. Soc., Providence, RI, 1993, 1-10.
  6. Abraham-Shrauner B., Leach P.G.L., Govinder K.S., Ratcliff G., Hidden and contact symmetries of ordinary differential equations, J. Phys. A: Math. Gen. 28 (1995), 6707-6716.
  7. Adam A.A., Mahomed F.M., Integration of ordinary differential equations via nonlocal symmetries, Nonlinear Dynam. 30 (2002), 267-275.
  8. Ballesteros Á., Enciso A., Herranz F.J., Ragnisco O., A maximally superintegrable system on an n-dimensional space of nonconstant curvature, Phys. D 237 (2008), 505-509, math-ph/0612080.
  9. Bhuvaneswari A., Kraenkel R.A., Senthilvelan M., Application of the λ-symmetries approach and time independent integral of the modified Emden equation, Nonlinear Anal. Real World Appl. 13 (2012), 1102-1114.
  10. Bhuvaneswari A., Kraenkel R.A., Senthilvelan M., Lie point symmetries and the time-independent integral of the damped harmonic oscillator, Phys. Scr. 83 (2011), 055005, 5 pages.
  11. Bruzon M.S., Gandarias M.L., Senthilvelan M., On the nonlocal symmetries of certain nonlinear oscillators and their general solution, Phys. Lett. A 375 (2011), 2985-2987.
  12. Catalano Ferraioli D., Nonlocal aspects of λ-symmetries and ODEs reduction, J. Phys. A: Math. Theor. 40 (2007), 5479-5489, math-ph/0702039.
  13. Catalano-Ferraioli D., Nonlocal interpretation of λ-symmetries, in Proceedings of the International Conference "Symmetry and Perturbation Theory" (Otranto, Italy, 2007), Editors G. Gaeta, S. Walcher, R. Vitolo, World Scientific, Singapore, 2008, 241-242.
  14. Catalano Ferraioli D., Morando P., Local and nonlocal solvable structures in the reduction of ODEs, J. Phys. A: Math. Theor. 42 (2009), 035210, 15 pages, arXiv:0807.3276.
  15. Cicogna G., Reduction of systems of first-order differential equations via Λ-symmetries, Phys. Lett. A 372 (2008), 3672-3677, arXiv:0802.3581.
  16. Cicogna G., Gaeta G., Noether theorem for μ-symmetries, J. Phys. A: Math. Theor. 40 (2007), 11899-11921, arXiv:0708.3144.
  17. Eisenhart L.P., Continuous groups of transformations, Dover Publications Inc., New York, 1961.
  18. Gaeta G., Morando P., On the geometry of lambda-symmetries and PDE reduction, J. Phys. A: Math. Gen. 37 (2004), 6955-6975, math-ph/0406051.
  19. Gandarias M.L., Nonlocal symmetries and reductions for some ordinary differential equations, Theoret. Math. Phys. 159 (2009), 779-786.
  20. Gandarias M.L., Bruzón M.S., Reductions for some ordinary differential equations through nonlocal symmetries, J. Nonlinear Math. Phys. 18 (2011), suppl. 1, 123-133.
  21. González-Gascón F., González-López A., Newtonian systems of differential equations, integrable via quadratures, with trivial group of point symmetries, Phys. Lett. A 129 (1988), 153-156.
  22. González-López A., Symmetry and integrability by quadratures of ordinary differential equations, Phys. Lett. A 133 (1988), 190-194.
  23. Govinder K.S., Leach P.G.L., A group-theoretic approach to a class of second-order ordinary differential equations not possessing Lie point symmetries, J. Phys. A: Math. Gen. 30 (1997), 2055-2068.
  24. Govinder K.S., Leach P.G.L., On the determination of non-local symmetries, J. Phys. A: Math. Gen. 28 (1995), 5349-5359.
  25. Levi D., Rodríguez M.A., λ-symmetries for discrete equations, J. Phys. A: Math. Theor. 43 (2010), 292001, 9 pages, arXiv:1004.4808.
  26. Mathews P.M., Lakshmanan M., On a unique nonlinear oscillator, Quart. Appl. Math. 32 (1974), 215-218.
  27. Morando P., Deformation of Lie derivative and μ-symmetries, J. Phys. A: Math. Theor. 40 (2007), 11547-11559.
  28. Muriel C., Romero J.L., C∞-symmetries and integrability of ordinary differential equations, in Proceedings of the I Colloquium on Lie Theory and Applications (Vigo, 2000), Colecc. Congr., Vol. 35, Univ. Vigo, Vigo, 2002, 143-150.
  29. Muriel C., Romero J.L., C∞-symmetries and non-solvable symmetry algebras, IMA J. Appl. Math. 66 (2001), 477-498.
  30. Muriel C., Romero J.L., C∞-symmetries and reduction of equations without Lie point symmetries, J. Lie Theory 13 (2003), 167-188.
  31. Muriel C., Romero J.L., First integrals, integrating factors and λ-symmetries of second-order differential equations, J. Phys. A: Math. Theor. 42 (2009), 365207, 17 pages.
  32. Muriel C., Romero J.L., Integrating factors and λ-symmetries, J. Nonlinear Math. Phys. 15 (2008), suppl. 3, 300-309.
  33. Muriel C., Romero J.L., λ-symmetries on the derivation of first integrals of ordinary differential equations, in Proceedings "WASCOM 2009" 15th Conference on Waves and Stability in Continuous Media, World Sci. Publ., Hackensack, NJ, 2010, 303-308.
  34. Muriel C., Romero J.L., New methods of reduction for ordinary differential equations, IMA J. Appl. Math. 66 (2001), 111-125.
  35. Muriel C., Romero J.L., Nonlocal transformations and linearization of second-order ordinary differential equations, J. Phys. A: Math. Theor. 43 (2010), 434025, 13 pages.
  36. Muriel C., Romero J.L., Prolongations of vector fields and the property of the existence of invariants obtained by differentiation, Theoret. Math. Phys. 133 (2002), 1565-1575.
  37. Muriel C., Romero J.L., Second-order ordinary differential equations and first integrals of the form A(t,x)x'+B(t,x), J. Nonlinear Math. Phys. 16 (2009), suppl. 1, 209-222.
  38. Muriel C., Romero J.L., Olver P.J., Variational C∞-symmetries and Euler-Lagrange equations, J. Differential Equations 222 (2006), 164-184.
  39. Nucci M.C., Lie symmetries of a Painlevé-type equation without Lie symmetries, J. Nonlinear Math. Phys. 15 (2008), 205-211.
  40. Nucci M.C., Leach P.G.L., The determination of nonlocal symmetries by the technique of reduction of order, J. Math. Anal. Appl. 251 (2000), 871-884.
  41. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1986.
  42. Ovsiannikov L.V., Group analysis of differential equations, Academic Press Inc., New York, 1982.
  43. Popovych R.O., Boyko V.M., Differential invariants and application to Riccati-type systems, in Proceedinds of Fourth International Conference "Symmetry in Nonlinear Mathematical Physics" (Kyiv, 2001), Proceedings of Institute of Mathematics, Kyiv, Vol. 43, Part 1, Editors A.G. Nikitin, V.M. Boyko, R.O. Popovych, Institute of Mathematics, Kyiv, 2002, 184-193, math-ph/0112057.
  44. Pucci E., Saccomandi G., On the reduction methods for ordinary differential equations, J. Phys. A: Math. Gen. 35 (2002), 6145-6155.
  45. Stephani H., Differential equations: their solution using symmetries, Cambridge University Press, Cambridge, 1989.
  46. Yasar E., Integrating factors and first integrals for Liénard type and frequency-damped oscillators, Math. Probl. Eng. 2011 (2011), 916437, 10 pages.

Previous article   Contents of Volume 8 (2012)