Interdependence Between the Laurent-Series and Elliptic Solutions of Nonintegrable Systems

  • S. Yu. Vernov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3718)


The standard methods for the search for the elliptic solutions consist of two independent steps: transformation of a nonlinear polynomial differential equation into a nonlinear algebraic system and the search for solutions of the obtained system. It has been demonstrated by the example of the generalized Hénon–Heiles system that the use of the Laurent-series solutions of the initial differential equation assists to solve the obtained algebraic system and, thereby, simplifies the search for elliptic solutions. This procedure has been automatized with the help of the computer algebra systems Maple and REDUCE. The Laurent-series solutions also assist to solve the inverse problem: to prove the non-existence of elliptic solutions. Using the Hone’s method based on the use the Laurent-series solutions and the residue theorem, we have proved that the cubic complex one-dimensional Ginzburg–Landau equation has neither elliptic standing wave nor elliptic travelling wave solutions. To find solutions of the initial differential equation in the form of the Laurent series we use the Painlevé test.


Elliptic Function Algebraic System Laurent Series Arbitrary Parameter Landau Equation 
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  1. 1.
    Weiss, J.: Bäcklund transformation and linearizations of the Hénon–Heiles system. Phys. Lett. A 102, 329–331 (1984); Bäcklund transformation and the Hénon–Heiles system. Phys. Lett. A 105, 387–389 (1984)Google Scholar
  2. 2.
    Santos, G.S.: Application of finite expansion in elliptic functions to solve differential equations. J. Phys. Soc. Japan 58, 4301–4310 (1989)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Conte, R., Musette, M.: Linearity inside nonlinearity: exact solutions to the complex Ginzburg–Landau equation. Phisica D 69, 1–17 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Timoshkova, E.I.: A New class of trajectories of motion in the Hénon–Heiles potential field. Astron. Zh. 76, 470–475 (1999) (in Russian); Astron. Rep. 43, 406–411 (1999) (in English)Google Scholar
  5. 5.
    Fan, E.: An algebraic method for finding a series of exact solutions to integrable and nonintegrable nonlinear evolutions equations. J. Phys. A 36, 7009–7026 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kudryashov, N.A.: Nonlinear differential equations with exact solutions expressed via the Weierstrass function, nlin.CD/0312035Google Scholar
  7. 7.
    Timoshkova, E.I., Vernov, S.Y.: On two nonintegrable cases of the generalized Hénon–Heiles system with an additional nonpolynomial term. math-ph/0402049. Yadernaya Fizika (Physics of Atomic Nuclei) 68(11) (2005) (in press)Google Scholar
  8. 8.
    Kudryashov, N.A.: Simplest equation method to look for exact solutions of nonlinear differential equations, nlin.SI/0406007Google Scholar
  9. 9.
    Musette, M., Conte, R.: Analytic solitary waves of nonintegrable equations. Physica D 181, 70–79 (2003); nlin.PS/0302051zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Vernov, S.Y.: Construction of solutions for the generalized Hénon–Heiles system with the help of the Painlevé test. TMF (Theor. Math. Phys.) 135, 409–419 (2003) (in Russian); 792–801 (in English), math-ph/0209063Google Scholar
  11. 11.
    Ginzburg, V.L., Landau, L.D.: On the theory of superconductors, Zh. Eksp. Teor. Fiz (Sov. Phys. JETP) 20, 1064–1082 (1950) (in Russian); In: Landau, L.D. (ed.), Collected Papers. Pergamon Press, Oxford, p. 546 (1950) (in English)Google Scholar
  12. 12.
    Hone, A.N.W.: Non-existence of elliptic travelling wave solutions of the complex Ginzburg–Landau equation. Physica D 205, 292–306 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Conte, R., Musette, M.: Solitary waves of nonlinear nonintegrable equations. nlin.PS/0407026Google Scholar
  14. 14.
    Vernov, S.Y.: Construction of single-valued solutions for nonintegrable systems with the help of the Painlevé test. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) Proc. Int. Conference Computer Algebra in Scientific Computing (CASC 2004), St. Petersburg, Russia, pp. 457–465. Technische Universitat, Munchen (2004); nlin.SI/0407062Google Scholar
  15. 15.
    Ablowitz, M.J., Ramani, A., Segur, H.: A connection between nonlinear evolution equations and ordinary differential equations of P-type. I & II. J. Math. Phys. 21, 715–721, 1006–1015 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hénon, M., Heiles, C.: The applicability of the third integral of motion: Some numerical experiments. Astronomical J. 69, 73–79 (1964)CrossRefGoogle Scholar
  17. 17.
    Davenport, J.H., Siret, Y., Tournier, E.: Calcul Formel, Systemes et Algorithmes de Manipulations Algebriques, Masson, Paris, New York (1987)Google Scholar
  18. 18.
    Hearn, A.C.: REDUCE. User’s Manual, Vers. 3.8,, REDUCE. User’s and Contributed Packages Manual, Vers. 3.7, CA and Codemist Ltd., Santa Monica, California (1999),
  19. 19.
    Heck, A.: Introduction to Maple, 3rd edn. Springer, New York (2003)zbMATHGoogle Scholar
  20. 20.
    van Hoeij, M.A.: package algcurves, Maple V and Maple 6,
  21. 21.
    Aranson, I., Kramer, L.: The world of the complex Ginzburg–Landau equation. Rev. Mod. Phys. 74, 99–143 (2002); cond-mat/0106115 CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    van Hecke, M., Storm, C., van Saarlos, W.: Sources, sinks and wavenumber selection in coupled CGL equations and experimental implications for counter-propagating wave systems. Phisica D 133, 1–47 (1999); Patt-sol/9902005Google Scholar
  23. 23.
    van Saarloos, W., Hohenberg, P.C.: Fronts, pulses, sources and sinks in generalized complex Ginzburg–Landau equations. Phisica D 56, 303–367 (1992); Erratum 69, p. 209 (1993)Google Scholar
  24. 24.
    Cariello, F., Tabor, M.: Painlevé expansions for nonintegrable evolution equations. Phisica D 39, 77–94 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Erdélyi, A., et al. (eds.): Higher Transcendental Functions (based, in part, on notes left by H. Bateman), vol. 3. MC Graw-Hill Book Company, New York (1955)Google Scholar
  26. 26.
    Vernov, S.Y.: On elliptic solutions of the cubic complex one-dimensional Ginzburg–Landau equation, nlin.PS/0503009Google Scholar
  27. 27.

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • S. Yu. Vernov
    • 1
  1. 1.Skobeltsyn Institute of Nuclear PhysicsMoscow State UniversityMoscowRussia

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