Skip to main content
SpringerLink
Log in

Special solutions of a high-order equation for waves in a liquid with gas bubbles

  • Published:
Regular and Chaotic Dynamics Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

A fifth-order nonlinear partial differential equation for the description of nonlinear waves in a liquid with gas bubbles is considered. Special solutions of this equation are studied. Some elliptic and simple periodic traveling wave solutions are constructed. Connection of self-similar solutions with Painlevé transcendents and their high-order analogs is discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ablowitz, M. J. and Clarkson, P.A., Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Math. Soc. Lecture Note Ser., vol. 149, Cambridge: Cambridge Univ. Press, 1991.

    Book  MATH  Google Scholar 

  2. Borisov, A.V. and Mamaev, I. S., Modern Methods of the Theory of Integrable Systems, Moscow: R&C Dynamics, ICS, 2003 (Russian).

    MATH  Google Scholar 

  3. Kudryashov, N.A., Methods of Nonlinear Mathematical Physics, Moscow: Intellekt, 2010 (Russian).

    Google Scholar 

  4. Polyanin, A.D. and Zaitsev, V. F., Handbook of Nonlinear Partial Differential Equations, Boca Raton, Fla.: Chapman and Hall/CRC, 2011.

    MATH  Google Scholar 

  5. Borisov, A. V. and Mamaev, I. S., Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere, Regul. Chaotic Dyn., 2013, vol. 18, no. 4, pp. 356–371.

    Article  MathSciNet  MATH  Google Scholar 

  6. Borisov, A. V. and Mamaev, I. S., The Dynamics of the Chaplygin Ball with a Fluid-Filled Cavity, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 490–496; see also: Rus. J. Nonlin. Dyn., 2012, vol. 8, no. 1, pp. 103–111.

    Article  MathSciNet  MATH  Google Scholar 

  7. Biswas, A. and Konar, S., Introduction to Non-Kerr Law Optical Solitons, Boca Raton, Fla.: Chapman and Hall/CRC, 2007.

    MATH  Google Scholar 

  8. Polyanin, A.D., On the Nonlinear Instability of the Solutions of Hydrodynamic-Type Systems, JETP Lett., 2009, vol. 90, no. 3, pp. 217–221; see also: Pis’ma v Zh. Èksper. Teoret. Fiz., 2009, vol. 90, no. 3, pp. 238–242.

    Article  Google Scholar 

  9. Vitanov, N.K., Jordanov, I.P., and Dimitrova, Z. I., On Nonlinear Population Waves, Appl. Math. Comput., 2009, vol. 215, no. 8, pp. 2950–2964.

    Article  MathSciNet  MATH  Google Scholar 

  10. Kudryashov, N.A. and Sinelshchikov, D. I., Extended Models of Non-Linear Waves in Liquid with Gas Bubbles, Internat. J. Non-Linear Mech., 2014, vol. 63, pp. 31–38.

    Article  Google Scholar 

  11. Kudryashov, N.A., Soukharev, M. B., and Demina, M.V., Elliptic Traveling Waves of the Olver Equation, Commun. Nonlinear Sci. Numer. Simul., 2012, vol. 17, no. 11, pp. 4104–4114.

    Article  MathSciNet  MATH  Google Scholar 

  12. Kudryashov, N.A., Higher Painlevé Transcendents As Special Solutions of Some Nonlinear Integrable Hierarchies, Regul. Chaotic Dyn., 2014, vol. 19, no. 1, pp. 48–63.

    Article  MathSciNet  Google Scholar 

  13. Borisov, A. V. and Kudryashov, N.A., Paul Painlevé and His Contribution to Science, Regul. Chaotic Dyn., 2014, vol. 19, no. 1, pp. 1–19.

    Article  MathSciNet  Google Scholar 

  14. Kaup, D. J., On the Inverse Scattering Problem for Cubic Eigenvalue Problems of the Class φ xxx + 6Qφ x + 6Rφ = λφ, Stud. Appl. Math., 1980, vol. 62, no. 3, pp. 189–216.

    MathSciNet  MATH  Google Scholar 

  15. Kupershmidt, B. A. and Wilson, G., Modifying Lax Equations and the Second Hamiltonian Structure, Invent. Math., 1981, vol. 62, no. 3, pp. 403–436.

    Article  MathSciNet  MATH  Google Scholar 

  16. Sawada, K. and Kotera, T., A Method for Finding N-Soliton Solutions of the K. d.V. Equation and K. d. V.-Like Equation, Progr. Theoret. Phys., 1974, vol. 51, pp. 1355–1367.

    Article  MathSciNet  MATH  Google Scholar 

  17. Kudryashov, N.A., Transcendents Defined by Nonlinear Fourth-Order Ordinary Differential Equations, J. Phys. A. Math. Gen., 1999, vol. 32, no. 6, pp. 999–1013.

    Article  MathSciNet  MATH  Google Scholar 

  18. Demina, M.V. and Kudryashov, N.A., Point Vortices and Polynomials of the Sawada — Kotera and Kaup-Kupershmidt Equations, Regul. Chaotic Dyn., 2011, vol. 16, no. 6, pp. 562–576.

    Article  MathSciNet  MATH  Google Scholar 

  19. Demina, M.V. and Kudryashov, N.A., From Laurent Series to Exact Meromorphic Solutions: The Kawahara Equation, Phys. Lett. A, 2010, vol. 374, no. 39, pp. 4023–4029.

    Article  MathSciNet  MATH  Google Scholar 

  20. Demina, M.V. and Kudryashov, N.A., Explicit Expressions for Meromorphic Solutions of Autonomous Nonlinear Ordinary Differential Equations, Commun. Nonlinear Sci. Numer. Simul., 2011, vol. 16, no. 3, pp. 1127–1134.

    Article  MathSciNet  MATH  Google Scholar 

  21. Kudryashov, N.A., Sinelshchikov, D. I., and Demina, M.V., Exact Solutions of the Generalized Bretherton Equation, Phys. Lett. A, 2011, vol. 375, no. 7, pp. 1074–1079.

    Article  MathSciNet  MATH  Google Scholar 

  22. Kudryashov, N.A. and Sinelshchikov, D. I., Exact Solutions of the Swift — Hohenberg Equation with Dispersion, Commun. Nonlinear Sci. Numer. Simul., 2012, vol. 17, no. 1, pp. 26–34.

    Article  MathSciNet  MATH  Google Scholar 

  23. Demina, M.V. and Kudryashov, N.A., Elliptic Solutions in the Hénon-Heiles Model, Commun. Nonlinear Sci. Numer. Simul., 2014, vol. 19, no. 3, pp. 471–482.

    Article  MathSciNet  Google Scholar 

  24. Cosgrove, Ch. M., Higher-Order Painlevé Equations in the Polynomial Class: 1. Bureau Symbol P2, Stud. Appl. Math., 2000, vol. 104, no. 1, pp. 1–65.

    Article  MathSciNet  MATH  Google Scholar 

  25. Kudryashov, N.A., Two Hierarchies of Ordinary Differential Equations and Their Properties, Phys. Lett. A, 1999, vol. 252, nos. 3–4, pp. 173–179.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe Shosse 31, 115409, Moscow, Russia

    Nikolay A. Kudryashov & Dmitry I. Sinelshchikov

Authors
  1. Nikolay A. Kudryashov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dmitry I. Sinelshchikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nikolay A. Kudryashov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kudryashov, N.A., Sinelshchikov, D.I. Special solutions of a high-order equation for waves in a liquid with gas bubbles. Regul. Chaot. Dyn. 19, 576–585 (2014). https://doi.org/10.1134/S1560354714050050

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1560354714050050

MSC2010 numbers

Keywords

Search

Navigation