Skip to main content
SpringerLink
Log in

Bifurcation and Symmetry in Problems of Capillarity-Gravity Waves

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We expose a general scheme for applying the methods of group analysis to the problems of branching theory with violation of symmetry. We consider application to construction and study of the branching equation in the problem of capillarity-gravity waves in a spatial layer of a floating fluid.

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. Yudovich V. I., “Free convection and branching,” Prikl. Mat. Mekh., 31, No. 1, 101-111 (1967).

    Google Scholar 

  2. Ovchinnikova S. N. and Yudovich V. I., “Calculation of the secondary steady flow between rotating cylinders,” Prikl. Mat. Mekh., 32, No. 5, 858-868 (1968).

    Google Scholar 

  3. Loginov B. V. and Trenogin V. A., “On use of group properties for defining multiparametric families of solutions to nonlinear equations,” Mat. Sb., 85, No. 3, 440-454 (1971).

    Google Scholar 

  4. Loginov B. V., The Branching Theory of Solutions to Nonlinear Equations Under the Conditions of Group Invariance [in Russian], Fan, Tashkent (1985).

    Google Scholar 

  5. Loginov B. V. and Trenogin V. A., “On use of group invariance in the branching theory,” Differentsial0nye Uravneniya, 11, No. 8, 1518-1521 (1975).

    Google Scholar 

  6. Ovsyannikov L. V., Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  7. Loginov B. V., Rakhmatova Kh. R., and Yuldashev N. N., “On construction of the branching equation from its symmetry group,” in: Equations of Mixed Types and Problems with a Free Boundary [in Russian], Fan, Tashkent, 1987, pp. 183-195.

    Google Scholar 

  8. Loginov B. V., “On the construction of the general form of branching equation by its group symmetry,” in: Equadiff-VII. Enlarged Abstracts, Praha, 1989, pp. 48-50.

  9. Loginov B. V., “Group analysis methods for constructions and investigation of the bifurcation equation,” Appl. Math., 37, No. 4, 241-248 (1992).

    Google Scholar 

  10. Loginov B. V., “Group analysis in the problems of branching theory with violation of symmetries,” in: Proceedings of the First International Conference “Differential Equations and Some of Their Applications,” Saransk, December 20-22, 1994, Saransk, 1995, pp. 103-119.

  11. Loginov B. V., “On application of vector invariants to determination of the general form of the branching equation under the conditions of group invariance,” Dokl. Akad. Nauk SSSR, 29, No. 5, 1045-1050 (1981).

    Google Scholar 

  12. Sattinger D. H., Group Theoretic Methods in Bifurcation Theory, Springer-Verlag, Berlin (1979). (Lecture Notes in Math; 762.)

    Google Scholar 

  13. Loginov B. V. and Rakhmatova Kh. R., “An application of the branching theory with group invariance to construction of branching periodic solutions of the problem of phase transitions in the statistic crystal theory,” in: Differential Equations and There Application in Mechanics [in Russian], Fan, Tashkent, 1985, pp. 114-136.

    Google Scholar 

  14. Yuldashev N. N., A Construction of the General Form of the Branching Equation from Its Symmetry Group [in Russian], Dis. Kand. Fiz.-Mat. Nauk, the V. I. Romanovskiľ Inst. Mat., Tashkent (1991).

    Google Scholar 

  15. Loginov B. V. and Kuznetsov A. O., “Capillary-gravity waves over the flat surface,” European J. Mech. B/Fluids, 15, No. 2, 259-280 (1996).

    Google Scholar 

  16. Loginov B. V. and Trofimov E. V., “The calculation of capillarity-gravity waves at the interface of two fluids,” in: Differential Equations of Mathematical Physics and Some of Their Applications, Fan, Tashkent, 1989, pp. 57-66.

    Google Scholar 

  17. Loginov B. V. and `Ergashbaev T., “Multidimensional branching and the problem of capillarity-gravity waves on a cylinder surface,” in: Problems of Computation and Applied Mathematics [in Russian], Fan, Tashkent, 1993, pp. 89-100.

    Google Scholar 

  18. Abdullaeva F. Dzh., Branching and Stability of Solutions to a System of Differential Equations Defining the Free Surface of a Magnetic Fluid [in Russian], Dis. Kand. Fiz.-Mat. Nauk, the V. I. Romanovskiľ Inst. Mat., Tashkent (1993).

    Google Scholar 

  19. Loginov B. V., “Bifurcation theory methods in the problem about capillarity-gravity waves in spatial layer of floating fluid,” in: Abstracts: International Congress of Mathematicians (August 18-27, 1998), p. 29.

  20. Loginov B. V. and Karpova S. A., “Branching and symmetry in the problem of capillarity-gravity waves on the surface of a floating fluid,” in: Abstracts: International Conference “Symmetry in Natural Sciences,” August 23-29, 1998, Inst.Vychisl. Modelirovaniya SO RAN, Krasnoyarsk, pp. 86-87.

    Google Scholar 

  21. Loginov B. V. and Grishina S. A., “The bifurcation problem of capillarity-gravity waves on the surface of a spatial layer of a floating fluid,” submitted to VINITI on 1998, No. 2456-B99.

  22. Nekrasov A. I., Exact Theory of Steady Waves Form on the Surface of a Heavy Fluid [in Russian], Akad. Nauk SSSR, Moscow (1951).

    Google Scholar 

  23. Kochin N. E., “Determination reregoreuse des ondes irrotationelles periodiques dans une canal profondeur finie,” Math.Ann., Bd 95, 595-634 (1928).

    Google Scholar 

  24. Kochin N. E., “Über den Einfluss des Bondenprofils auf die Wellen an der Grenzflache von zwei Flüssigkeiten verschiedener Dichten,” Izv. Akad, Nauk SSSR OMEN Ser. Geogr. Geofiz., No. 3, 357-381 (1937).

  25. Agranovich M. S., “Elliptic operators on closed manifolds,” in: Contemporary Problems of Mathematics. Fundamental Trends [in Russian], VINITI, Moscow, 1990, 63, pp. 5-129.

    Google Scholar 

  26. Loginov B. V., “Branching of solutions to nonlinear equations and group symmetry,” Vestnik Samarsk. Univ., No. 4, 15-70 (1998).

  27. Vaľnberg M. M. and Trenogin V. A., The Branching Theory of Solutions to Nonlinear Equations [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  28. Heine V., Group Theory in Quantum Mechanics [Russian translation], Izdat. Inostr. Lit., Moscow (1969).

    Google Scholar 

  29. Kittel Ch., Introduction to Solid State Physics [Russian translation], Nauka, Moscow (1978).

    Google Scholar 

  30. Lyubarskiľ G. Ya., Group Theory and Its Application to Physics [in Russian], GITTL, Moscow (1958).

    Google Scholar 

  31. Delone B. N., Alexandrov A. D., and Padurov N. N., Mathematical Fundamentals of Structure Analysis of Cristals [in Russian], ONTI, Leningrad and Moscow (1934).

    Google Scholar 

  32. Dubrovin B. A., Novikov S. P., and Fomenko A. T., Modern Geometry. Part 1 [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  33. Zhelobenko D. P., Compact Lie Groups and Their Representations [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  34. Naľmark M. A., Representation Theory of Groups [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  35. Loginov B. V., “Invariants and invariant solutions in the branching theory,” in: Direct and Inverse Problems for Partial Differential Equations and Some of Their Applications [in Russian], Fan, Tashkent, 1978, pp. 117-133.

    Google Scholar 

  36. Loginov B. V., “On invariant solutions in the branching theory,” Dokl. Akad. Nauk SSSR, 247, No. 5, 1048-1051 (1979).

    Google Scholar 

  37. Vladimirov S. A., “Ordinary differential equations with a discrete symmetry group,” Differentsial0nye Uravneniya, 12, No. 7, 1180-1189 (1975).

    Google Scholar 

  38. Rabinovitz P., “Some aspects of nonlinear eigenvalue problems,” Rocky Mountain J. Math., 12, 161-202 (1973).

    Google Scholar 

  39. Gabov S. A., “On existence of steady waves of finite amplitude on the surface of a floating fluid,” Zh. Vychisl. Matematiki i Mat. Fiziki, 28, No. 10, 1507-1519 (1988).

    Google Scholar 

  40. Gabov S. A. and Sveshnikov A. G., “Mathematical problems of dynamics of a floating fluid,” Mathematical Analysis [in Russian], VINITI, Moscow, 1990, pp. 3-86. (Itogi Nauki i Tekhniki, 28.)

    Google Scholar 

  41. Naľmark M. A., Linear Differential Operators [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  42. Loginov B. V. and Karpova S. A., “Calculation of asymptotics of periodic solutions of the problem of capillarity-gravity waves in a spatial layer of a floating fluid,” Vestnik Samarsk. Univ., No. 4, 69-70 (1997).

Download references

Author information

Authors and Affiliations

  1. Ul'yanovsk State Technical University, Ul'yanovsk

    B. V. Loginov

Authors
  1. B. V. Loginov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Loginov, B.V. Bifurcation and Symmetry in Problems of Capillarity-Gravity Waves. Siberian Mathematical Journal 42, 731–747 (2001). https://doi.org/10.1023/A:1010449615704

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1010449615704

Keywords

Search

Navigation