Advertisement

Theoretical and Mathematical Physics

, Volume 196, Issue 3, pp 1307–1319 | Cite as

Conformally Invariant Elliptic Liouville Equation and Its Symmetry-Preserving Discretization

  • D. Levi
  • L. Martina
  • P. Winternitz
Article
  • 17 Downloads

Abstract

The symmetry algebra of the real elliptic Liouville equation is an infinite-dimensional loop algebra with the simple Lie algebra o(3, 1) as its maximal finite-dimensional subalgebra. The entire algebra generates the conformal group of the Euclidean plane E2. This infinite-dimensional algebra distinguishes the elliptic Liouville equation from the hyperbolic one with its symmetry algebra that is the direct sum of two Virasoro algebras. Following a previously developed discretization procedure, we present a difference scheme that is invariant under the group O(3, 1) and has the elliptic Liouville equation in polar coordinates as its continuous limit. The lattice is a solution of an equation invariant under O(3, 1) and is itself invariant under a subgroup of O(3, 1), namely, the O(2) rotations of the Euclidean plane.

Keywords

Lie group partial differential equation discretization procedure 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    B. A. Dubrovin, S. P. Novikov and A. T. Fomenko, Modern Geometry: Methods and Applications [in Russian], Nauka, Moscow (1986); English transl.: Modern Geometry Methods and Applications: Part I. The Geometry of Surfaces, Transformation Groups, and Fields, Springer, Berlin (1992).CrossRefzbMATHGoogle Scholar
  2. 2.
    A. A. Belavin, A. N. Polyakov, and A. B. Zamolodchikov, “Infinite conformal symmetry group in two-dimensional quantum field theory,” Nucl. Phys. B, 241, 333–380 (1984).ADSCrossRefzbMATHGoogle Scholar
  3. 3.
    A. M. Polyakov, “Quantum geometry of bosonic strings,” Phys. Lett. B, 103, 207–210 (1981).ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    L. D. Faddeev and L. A. Takhtajan, “Liouville model on the lattice,” in: Field Theory, Quantum Gravity, and Strings (Lect. Notes Phys., Vol. 246, H. J. de Vega and N. Sánchez, eds.), Springer, Berlin (1986), pp. 166–179.Google Scholar
  5. 5.
    A. M. Polyakov, Gauge Fields and Strings [in Russian], RKhD, Izhevsk (1999); English transl. prev. ed. (Contemp. Concepts Phys., Vol. 3), Harwood Academic, New York (1987).zbMATHGoogle Scholar
  6. 6.
    H. Dorn and H.-J. Otto, “On correlation functions for non-critical strings with c ≤ 1 but d ≥ 1,” Phys. Lett. B, 291, 39–43 (1992).ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    A. B. Zamolodchikov and Al. B. Zamolodchikov, “Conformal bootstrap in Liouville field theory,” Nucl. Phys. B, 477, 577–605 (1996).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    J. Teschner, “Liouville theory revisited,” Class. Q. Grav., 18, R153–R222 (2001).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Yu. Nakayama, “Liouville field theory: A decade after the revolution,” Internat. J. Modern Phys. A, 19, 2771–2930 (2004).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    R. Jackiw, “Weyl symmetry and the Liouville theory,” Theor. Math. Phys., 148, 941–947 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A. Jaffe and C. H. Taubes, Vortices and Monopoles: Structure of Static Gauge Theories (Progr. Phys., Vol. 2), Birkhäuser, Boston (1980).zbMATHGoogle Scholar
  12. 12.
    D. Bartolucci and G. Tarantello, “Asymptotic blow-up analysis for singular Liouville type equations with applications,” J. Differ. Equ., 262, 3887–3931 (2017).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    G. P. Jorijadze, A. K. Pogrebkov, M. C. Polivanov, and S. V. Talalov, “Liouville field theory: IST and Poisson bracket structure,” J. Phys. A: Math. Gen., 19, 121–140 (1986).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    D. G. Crowdy, “General solutions to the 2D Liouville equation,” Internat. J. Eng. Sci., 35, 141–149 (1997).CrossRefzbMATHGoogle Scholar
  15. 15.
    A. V. Kiselev, “On the geometry of Liouville equation: Symmetries, conservation laws, and Bäcklund transformations,” Acta Math. Appl., 72, 33–40 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    D. Levi, L. Martina, and P. Winternitz, “Lie-point symmetries of the discrete Liouville equation,” J. Phys. A: Math. Theor., 48, 025204 (2015).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    D. Levi, L. Martina, and P. Winternitz, “Structure preserving discretizations of the Liouville equation and their numerical tests,” SIGMA, 11, 080 (2015).MathSciNetzbMATHGoogle Scholar
  18. 18.
    V. A. Dorodnitsyn, Applications of Lie Groups to Difference Equations, CRC Press, Boca Raton, Fla. (2011).zbMATHGoogle Scholar
  19. 19.
    D. Levi and P. Winternitz, “Continuous symmetries of difference equations,” J. Phys. A.: Math. Gen., 39, R1–R63 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    R. Rebelo and F. Valiquette, “Invariant discretization of partial differential equations admitting infinitedimensional symmetry groups,” J. Differ. Equ. Appl., 21, 285–318 (2015).CrossRefzbMATHGoogle Scholar
  21. 21.
    R. Rebelo and F. Valiquette, “Symmetry preserving numerical schemes for partial differential equations and their numerical tests,” J. Differ. Equ. Appl., 19, 737–757 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    M. A. Rodriguez and P. Winternitz, “Lie symmetries and exact solutions of first-order difference schemes,” J. Phys. A: Math. Gen., 37, 6129–6142 (2004).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    G. W. Bluman and S. Kumei, Symmetries and Differential Equations (Appl. Math. Sci., Vol. 81), Springer, New York (1989).CrossRefzbMATHGoogle Scholar
  24. 24.
    G. Gubbiotti, D. Levi, and C. Scimiterna, “On partial differential and difference equations with symmetries depending on arbitrary functions,” Acta Polytech., 56, 193–201 (2016).CrossRefGoogle Scholar
  25. 25.
    B. Champagne and P. Winternitz, “On the infinite-dimensional symmetry group of the Davey–Stewartson equations,” J. Math. Phys., 29, 1–8 (1988).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    D. David, N. Kamran, D. Levi, and P. Winternitz, “Subalgebras of loop algebras and symmetries of the Kadomtsev–Petviashvili equation,” Phys. Rev. Lett., 55, 2111–2113 (1985).ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    D. David, N. Kamran, D. Levi, and P. Winternitz, “Symmetry reduction for the Kadomtsev–Petviashvili equation using a loop algebra,” J. Math. Phys., 27, 1225–1237 (1986).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    D. David, D. Levi, and P. Winternitz, “Equations invariant under the symmetry group of the Kadomtsev–Petviashvili equation,” Phys. Lett. A, 129, 161–164 (1988).ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    L. Martina and P. Winternitz, “Analysis and applications of the symmetry group of the multidimensional three wave resonant interaction problem,” Ann. Phys., 196, 231–277 (1989).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    A. Yu. Orlov and P. Winternitz, “Algebra of pseudodifferential operators and symmetries of equations in the Kadomtsev–Petviashvili hierarchy,” J. Math. Phys., 38, 4644–4674 (1997).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    G. Paquin and P. Winternitz, “Group theoretical analysis of dispersive long wave equations in two space dimensions,” Phys. D, 46, 122–138 (1990).MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    S. Lie, “General theory of partial differential equations of an arbitrary order,” in: Lie Group Analysis: Classical Heritage (N. H. Ibragimov, ed.), ALGA Publ. (Blekinge Institute of Technology), Karlskrona, Sweden (2004), pp. 1–64.Google Scholar
  33. 33.
    U. Amaldi, “Sulla classificazione dei gruppi continui di trasformazioni di contatto dello spazio,” Mem. Soc. It. Scienze (ser. 3), 20, 167–350 (1918).Google Scholar
  34. 34.
    P. Medolaghi, “Sulla teoria dei gruppi infiniti continui,” Annali di Matematica, 25, 179–217 (1887).CrossRefzbMATHGoogle Scholar
  35. 35.
    V. A. Dorodnitsyn, R. Kozlov, S. V. Meleshko, and P. Winternitz, “Lie group classification of first-order delay ordinary differential equations,” J. Phys. A: Math. Theor., 51, 205202 (2018).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    M. I. Bakirova, V. A. Dorodnitsyn, and R. V. Kozlov, “Symmetry-preserving discrete schemes for some heat transfer equations,” J. Phys. A: Math. Gen., 30, 8139–8155 (1997).ADSCrossRefzbMATHGoogle Scholar
  37. 37.
    A. Bihlo, X. Coiteux-Roy, and P. Winternitz, “The Korteweg–de Vries equation and its symmetry-preserving discretization,” J. Phys. A: Math. Theor., 48, 055201 (2015).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    C. Budd and V. A. Dorodnitsyn, “Symmetry-adapted moving mesh schemes for the nonlinear Schrödinger equation,” J. Phys. A: Math. Gen., 34, 10387–10400 (2001).ADSCrossRefzbMATHGoogle Scholar
  39. 39.
    V. A. Dorodnitsyn and R. Kozlov, “A heat transfer with a source: The complete set of invariant difference schemes,” J. Nonlinear Math. Phys., 10, 16–50 (2003).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    D. Levi, S. Tremblay, and P. Winternitz, “Lie symmetries of multidimensional difference equations,” J. Phys. A: Gen. Math., 34, 9507–9524 (2001).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    D. Levi, P. Tempesta, and P. Winternitz, “Lorentz and Galilei invariance on lattices,” Phys. Rev. D, 69, 105011 (2004).ADSCrossRefGoogle Scholar
  42. 42.
    E. Hairer, G. Wanner, and C. Lubich, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (Springer Ser. Comput. Math., Vol. 31), Springer, Berlin (2006).zbMATHGoogle Scholar
  43. 43.
    A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge Univ. Press, Cambridge (2008).CrossRefzbMATHGoogle Scholar
  44. 44.
    J. E. Marsden and M. West, “Discrete mechanics and variational integrators,” Acta Numer., 10, 357–514 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    R. I. McLachlan and G. R. W. Quispel, “Geometric integrators for ODEs,” J. Phys. A: Math. Gen., 39, 5251–5286 (2006).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    V. A. Dorodnitsyn, R. Kozlov, and P. Winternitz, “Lie group classification of second-order ordinary difference equations,” J. Math. Phys., 41, 480–504 (2000).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    V. A. Dorodnitsyn, “Noether-type theorems for difference equations,” Appl. Numer. Math., 39, 307–321 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    V. A. Dorodnitsyn, R. Kozlov, and P. Winternitz, “Continuous symmetries of Lagrangians and exact solutions of discrete equations,” J. Math. Phys., 45, 336–359 (2004).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    D. Levi, S. Tremblay, and P. Winternitz, “Lie point symmetries of difference equations and lattices,” J. Phys. A: Math. Gen., 33, 8507–8523 (2000).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    R. Campoamor-Stursberg, M. A. Rodriguez, and P. Winternitz, “Symmetry preserving discretization of ordinary differential equations: Large symmetry groups and higher order equations,” J. Phys. A: Math. Theor., 49, 035201 (2016).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    V. A. Dorodnitsyn, E. Kaptsov, R. Kozlov, and P. Winternitz, “The adjoint equation method for constructing first integrals of difference equations,” J. Phys. A: Math. Theor., 48, 055202 (2015).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    R. Rebelo and P. Winternitz, “Invariant difference schemes and their application to sl(2,R) invariant ordinary differential equations,” J. Phys. A: Math. Theor., 42, 454016 (2009).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    A. Bourlioux, C. Cyr-Gagnon, and P. Winternitz, “Difference schemes with point symmetries and their numerical tests,” J. Phys. A: Math. Gen., 39, 6877–6896 (2006).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    V. E. Adler and S. Ya. Startsev, “Discrete analogues of the Liouville equation,” Theor. Math. Phys., 121, 1484–1495 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    A. Bihlo and F. Valiquette, “Symmetry-preserving numerical schemes,” in: Symmetries and Integrability of Difference Equations (D. Levi, R. Rebelo, and P. Winternitz, eds.), Springer, New York (2017), pp. 261–324; arXiv:1608.02557v2 [math.NA] (2016).CrossRefGoogle Scholar
  56. 56.
    G. Cicogna, “Symmetry classification of quasi-linear PDE’s containing arbitrary functions,” Nonlinear Dynam., 51, 309–316 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    W. I. Fushchych and N. I. Sedov, “The symmetry and some exact solutions of the nonlinear many-dimensional Liouville, d’Alembert, and eikonal equations,” J. Phys. A: Math. Gen., 16, 3645–3658 (1983).ADSMathSciNetCrossRefGoogle Scholar
  58. 58.
    R. Buckmire, “Application of a Mickens finite-difference scheme to the cylindrical Bratu–Gelfand problem,” Numer. Methods Partial Diff. Equ., 20, 327–337 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore (1994).zbMATHGoogle Scholar
  60. 60.
    P. J. Olver, “A survey of moving frames,” in: Computer Algebra and Geometric Algebra with Applications (Lect. Notes Computer Sci., Vol. 3519, H. Li, P. J. Olver, and G. Sommer, eds.), Springer, New York (2005), pp. 105–138.CrossRefGoogle Scholar
  61. 61.
    P. J. Olver, “On multivariate interpolation,” Stud. Appl. Math., 116, 201–240 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    P. J. Olver and J. Pohjanpelto, “Moving frames for Lie pseudo-groups,” Canad. J. Math., 60, 1336–1386 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    P. J. Olver, J. Pohjanpelto, and F. Valiquette, “On the structure of Lie pseudo-groups,” SIGMA, 5, 077 (2009).MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Dipartimento di Matematica e FisicaUniversità degli Studi Roma TreRomeItaly
  2. 2.Instituto Nazionale di Fisica NucleareSezione di Roma TreRomeItaly
  3. 3.Dipartimento di Matematica e FisicaUniversità del SalentoLecceItaly
  4. 4.Instituto Nazionale di Fisica NucleareSezione di LecceLecceItaly
  5. 5.Département de Mathématiques et de Statistique and Centre de Recherches MathématiquesUniversité de MontréalMontréalCanada

Personalised recommendations