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

SIGMA 12 (2016), 077, 18 pages      arXiv:1511.03025      https://doi.org/10.3842/SIGMA.2016.077

Solvable Structures Associated to the Nonsolvable Symmetry Algebra $\mathfrak{sl}(2,\mathbb{R})$

Adrián Ruiz and Concepción Muriel
Department of Mathematics, University of Cádiz, 11510 Puerto Real, Spain

Received November 11, 2015, in final form August 03, 2016; Published online August 08, 2016

Abstract
Third-order ordinary differential equations with Lie symmetry algebras isomorphic to the nonsolvable algebra $\mathfrak{sl}(2,\mathbb{R})$ admit solvable structures. These solvable structures can be constructed by using the basis elements of these algebras. Once the solvable structures are known, the given equation can be integrated by quadratures as in the case of solvable symmetry algebras.

Key words: first integral; solvable structure; $\mathcal{C}^{\infty}$-symmetry; nonsolvable algebra.

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References

  1. 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.
  2. 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.
  3. Adam A.A., Mahomed F.M., Integration of ordinary differential equations via nonlocal symmetries, Nonlinear Dynam. 30 (2002), 267-275.
  4. Barco M.A., Prince G.E., Solvable symmetry structures in differential form applications, Acta Appl. Math. 66 (2001), 89-121.
  5. Basarab-Horwath P., Integrability by quadratures for systems of involutive vector fields, Ukr. Math. J. 43 (1991), 1236-1242.
  6. Clarkson P.A., Olver P.J., Symmetry and the Chazy equation, J. Differential Equations 124 (1996), 225-246.
  7. Gat O., Symmetry algebras of third-order ordinary differential equations, J. Math. Phys. 33 (1992), 2966-2971.
  8. González-López A., Symmetry and integrability by quadratures of ordinary differential equations, Phys. Lett. A 133 (1988), 190-194.
  9. González-López A., Kamran N., Olver P.J., Lie algebras of vector fields in the real plane, Proc. London Math. Soc. 64 (1992), 339-368.
  10. Govinder K.S., Leach P.G.L., On the determination of non-local symmetries, J. Phys. A: Math. Gen. 28 (1995), 5349-5359.
  11. 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.
  12. Gusyatnikova V.N., Yumaguzhin V.A., Contact transformations and local reducibility of ODE to the form $y'''=0$, Acta Appl. Math. 56 (1999), 155-179.
  13. Hartl T., Athorne C., Solvable structures and hidden symmetries, J. Phys. A: Math. Gen. 27 (1994), 3463-3474.
  14. Hydon P.E., Symmetry methods for differential equations: a beginner's guide, Cambridge Texts in Applied Mathematics, Vol. 22, Cambridge University Press, Cambridge, 2000.
  15. Ibragimov N.H., A practical course in differential equations and mathematical modelling: classical and new methods, nonlinear mathematical models, symmetry and invariance principles, ALGA Publications, 2004.
  16. Ibragimov N.H., Nucci M.C., Integration of third order ordinary differential equations by Lie's method: equations admitting three-dimensional Lie algebras, Lie Groups Appl. 1 (1994), 49-64.
  17. Lie S., Classification und Integration von gewöhnlichen Differentialgleichungen zwischen $xy$, die eine Gruppe von Transformationen gestatten, Math. Ann. 32 (1888), 213-281.
  18. Mahomed F.M., Leach P.G.L., Normal forms for third order equations, in Finite-Dimensional Integrable Nonlinear Dynamical Systems (Johannesburg, 1988), World Sci. Publishing, Singapore, 1988, 178-189.
  19. Muriel C., Romero J.L., New methods of reduction for ordinary differential equations, IMA J. Appl. Math. 66 (2001), 111-125.
  20. Muriel C., Romero J.L., $C^\infty$-symmetries and non-solvable symmetry algebras, IMA J. Appl. Math. 66 (2001), 477-498.
  21. Muriel C., Romero J.L., First integrals, integrating factors and $\lambda$-symmetries of second-order differential equations, J. Phys. A: Math. Theor. 42 (2009), 365207, 17 pages.
  22. Muriel C., Romero J.L., Ruiz A., Solvable structures and $\mathcal{C}^{\infty}$-symmetries of second-order ordinary differential equations, arXiv:1605.07814.
  23. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, 2nd ed., Springer-Verlag, New York, 1993.
  24. Olver P.J., Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995.
  25. Ruiz A., Muriel C., Solvable structures and integrability by quadratures of third-order ordinary differential equations admitting the nonsolvable symmetry algebra $\mathfrak{sl}(2,\mathbb{R})$, in Proceedings of the XXIV Congress on Differential Equations and Applications / XIV Congress on Applied Mathematics (Cádiz, 2015), Servicio de publicaciones de la Universidad de Cádiz, 2015, 493-500.
  26. Ruiz A., Muriel C., Applications of $\mathcal{C}^{\infty}$-symmetries in the construction of solvable structures, in Trends in Differential Equations and Applications, SEMA SIMAI Springer Series, Vol. 8, Springer, Berlin, 2016, 387-403.
  27. Sardanashvily G., Advanced differential geometry for theoreticians: fiber bundles, jet manifolds and Lagrangian theory, LAP Lambert Academic Publishing, 2013.
  28. Schmucker A., Czichowski G., Symmetry algebras and normal forms of third order ordinary differential equations, J. Lie Theory 8 (1998), 129-137.
  29. Schwarz F., Algorithmic Lie theory for solving ordinary differential equations, Pure and Applied Mathematics (Boca Raton), Vol. 291, Chapman & Hall/CRC, Boca Raton, FL, 2008.
  30. Sherring J., Prince G., Geometric aspects of reduction of order, Trans. Amer. Math. Soc. 334 (1992), 433-453.
  31. Stephani H., Differential equations: their solution using symmetries, Cambridge University Press, Cambridge, 1989.

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