Skip to main content
SpringerLink
Log in

Periodic Oscillations and Waves in Nonlinear Weakly Coupled Dispersive Systems

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

Bifurcations of periodic solutions in autonomous nonlinear systems of weakly coupled equations are studied. A comparative analysis is carried out between the mechanisms of Lyapunov–Schmidt reduction of bifurcation equations for solutions close to harmonic oscillations and cnoidal waves. Sufficient conditions for the branching of orbits of solutions are formulated in terms of the Pontryagin functional depending on perturbing terms.

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. A. D. Bryuno, “The normal form of a system close to a Hamiltonian system,” Mat. Zametki 48 (5), 35–46 (1990) [Math. Notes 48, 1100–1108 (1990)].

    MathSciNet  MATH  Google Scholar 

  2. J. A. Gear and R. Grimshaw, “Weak and strong interactions between internal solitary waves,” Stud. Appl. Math. 70 (3), 235–258 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  3. A. T. Il’ichev, Solitary Waves in Hydromechanical Models (Fizmatlit, Moscow, 2003) [in Russian].

    Google Scholar 

  4. L. G. Kurakin and V. I. Yudovich, “Application of the Lyapunov–Schmidtmethod to the problem of the branching of a cycle from a family of equilibria in a system with multicosymmetry,” Sib. Mat. Zh. 41 (1), 136–149 (2000) [Sib. Math. J. 41, 114–124 (2000)].

    Article  MATH  Google Scholar 

  5. W. Littman, “On the existence of periodic waves near critical speed,” Commun. Pure Appl. Math. 10 (2), 241–269 (1957).

    Article  MathSciNet  MATH  Google Scholar 

  6. B. V. Loginov, Branching Theory for the Solutions of Nonlinear Equations under Conditions of Group Invariance (Fan, Tashkent, 1985) [in Russian].

    MATH  Google Scholar 

  7. B. V. Loginov, I. V. Konopleva, and Yu. B. Rusak, “Symmetry and potentiality in the general problem of the branching theory,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 4, 30–40 (2006) [Russ. Math. 50 (4), 28–38 (2006)].

    MATH  Google Scholar 

  8. B. V. Loginov and V. A. Trenogin, “The use of group properties to determine multi-parameter families of solutions of nonlinear equations,” Mat. Sb. 85 (3), 440–454 (1971) [Math. USSR, Sb. 14 (3), 438–452 (1971)].

    MathSciNet  MATH  Google Scholar 

  9. N. I. Makarenko, “On the bifurcation of solutions to invariant variational equations,” Dokl. Akad. Nauk 348 (3), 302–304 (1996) [Dokl. Math. 53 (3), 369–371 (1996)].

    MathSciNet  MATH  Google Scholar 

  10. I. G. Malkin, The Methods of Lyapunov and Poincaré in the Theory of Nonlinear Oscillations (URSS, Moscow, 2004) [in Russian].

    Google Scholar 

  11. V. K. Mel’nikov, “On the stability of the center for time-periodic perturbations,” Tr. Mosk. Mat. Obshch. 12, 3–52 (1963) [Trans. Moscow Math. Soc. 12, 1–56 (1963)].

    MathSciNet  MATH  Google Scholar 

  12. A. Mielke, Hamiltonian and Lagrangian Flows on Center Manifolds: With Applications to Elliptic Variational Problems (Springer, Berlin, 1991), Lect. Notes Math. 1489.

    Book  MATH  Google Scholar 

  13. L. S. Pontryagin, “On dynamical systems close to Hamiltonian systems,” Zh. Eksp. Teor. Fiz. 4 (9), 883–885 (1934).

    Google Scholar 

  14. A. M. Ter-Krikorov, “The existence of periodic waves which degenerate into a solitary wave,” Prikl. Mat. Mekh. 24 (4), 622–636 (1960) [J. Appl. Math. Mech. 24, 930–949 (1961)].

    MATH  Google Scholar 

  15. M. M. Vainberg and V. A. Trenogin, Theory of Branching of Solutions of Non-linear Equations (Nauka, Moscow, 1969; Noordhoff Int. Publ., Leyden, 1974).

    MATH  Google Scholar 

  16. A. Vanderbauwhede, “Lyapunov–Schmidt method for dynamical systems,” in Encyclopedia of Complexity and Systems Science (Springer, New York, 2009), pp. 5299–5315.

    Chapter  Google Scholar 

  17. J. D. Wright and A. Scheel, “Solitary waves and their linear stability in weakly coupled KdV equations,” Z. Angew. Math. Phys. 58 (4), 535–570 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  18. V. I. Yudovich, “Cosymmetry, degeneration of solutions of operator equations, and onset of a filtration convection,” Mat. Zametki 49 (5), 142–148 (1991) [Math. Notes 49, 540–545 (1991)].

    MathSciNet  MATH  Google Scholar 

  19. V. I. Yudovich, “Implicit function theorem for cosymmetric equations,” Mat. Zametki 60 (2), 313–317 (1996) [Math. Notes 60, 235–238 (1996)].

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Novosibirsk State University, ul. Pirogova 1, Novosibirsk, 630090, Russia

    N. I. Makarenko & Z. V. Makridin

  2. Lavrentyev Institute of Hydrodynamics, Siberian Branch of the Russian Academy of Sciences, pr. Lavrent’eva 15, Novosibirsk, 630090, Russia

    N. I. Makarenko & Z. V. Makridin

Authors
  1. N. I. Makarenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Z. V. Makridin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. I. Makarenko.

Additional information

Original Russian Text © N.I. Makarenko, Z.V. Makridin, 2018, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2018, Vol. 300, pp. 158–167.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Makarenko, N.I., Makridin, Z.V. Periodic Oscillations and Waves in Nonlinear Weakly Coupled Dispersive Systems. Proc. Steklov Inst. Math. 300, 149–158 (2018). https://doi.org/10.1134/S0081543818010121

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543818010121

Search

Navigation