Advanced Search
Article Contents
Article Contents

Generalized pitchfork bifurcation on a two-dimensional gaseous star with self-gravity and surface tension

Abstract Related Papers Cited by
  • Travelling wave solutions of a two-dimensional gaseous star with self-gravity and surface tension are considered. The star rotates along a clockwise direction with a travelling speed. The governing equations on a whole unknown domain are changed to ones on the boundary using the Dirichlet-Neumann operator. The problem of the existence of its periodic solutions is equivalent to one of a functional equation. After applying the method of Lyapunov-Schmidt reduction, the reduced equation of this functional equation has a generalized Pitchfork bifurcation with the bifurcation parameter being the travelling speed. This shows that there exist two nontrivial periodic solutions.
    Mathematics Subject Classification: Primary: 35B10, 35B32; Secondary: 76B15.


    \begin{equation} \\ \end{equation}
  • [1]

    C. J. Amick and K. Kirchgässner, A theory of solitary water-waves in the presence of surface tension, Arch. Rat. Mech. Anal., 105 (1989), 1-49.doi: 10.1007/BF00251596.


    C. J. Amick and J. F. Toland, On periodic water-waves and their convergence to solitary waves in the long-wave limit, Philos. Trans. Roy. Soc. London Ser. A, 303 (1981), 633-669.doi: 10.1098/rsta.1981.0231.


    J. T. Beale, Exact solitary water waves with capillary ripples at infinity, Comm. Pure Appl. Math., 44 (1991), 211-257.doi: 10.1002/cpa.3160440204.


    B. Buffoni, E. N. Dancer and J. F. Toland, The sub-harmonic bifurcation of Stokes waves, Arch. Rational Mech. Anal., 153 (2000), 241-271.doi: 10.1007/s002050000087.


    B. Buffoni and M. D. Groves, A multiplicity result for solitary gravity-capillary waves in deep water via critical-point theory, Arch. Ration. Mech. Anal., 146 (1999), 183-220.doi: 10.1007/s002050050141.


    B. Buffoni, M. D. Groves and J. F. Toland, A plethora of solitary gravity-capillary water waves with nearly critical Bond and Froude numbers, Philos. Trans. Roy. Soc. London Ser. A, 354 (1996), 575-607.doi: 10.1098/rsta.1996.0020.


    A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527.doi: 10.1002/cpa.3046.


    W. Craig and D. P. Nicholls, Travelling two and three dimensional capillary gravity water waves, SIAM J. Math. Anal., 32 (2000), 323-359.doi: 10.1137/S0036141099354181.


    W. Craig and D. P. Nicholls, Traveling gravity water waves in two and three dimensions, EJMB/Fluids, 21 (2002), 615-641.doi: 10.1016/S0997-7546(02)01207-4.


    S. Deng and S. Sun, Exact theory of three-dimensional water waves at the critical speed, SIAM J. Math. Anal., 42 (2010), 2721-2761.doi: 10.1137/09077922X.


    S. Deng, Two-dimensional travelling waves over a moving bottom, J. Diff. Equa., 248 (2010), 1777-1793.doi: 10.1016/j.jde.2009.09.005.


    S. Deng and S. Sun, Three-dimensional gravity-capillary waves on water-small surface tension case, Phys. D, 238 (2009), 1735-1751.doi: 10.1016/j.physd.2009.05.012.


    F. Dias and C. Kharif, Nonlinear gravity and capillary-gravity waves, Annu. Rev. Fluid Mech., 31 (1999), 301-346.doi: 10.1146/annurev.fluid.31.1.301.


    M. D. Groves, An existence theory for three-dimensional periodic travelling gravity-capillary water waves with bounded transverse profiles, Phys. D, 152/153 (2001), 395-415.doi: 10.1016/S0167-2789(01)00182-8.


    M. D. Groves, Three-dimensional travelling gravity-capillary water waves, GAMM-Mitt., 30 (2007), 8-43.doi: 10.1002/gamm.200790013.


    M. D. Groves and M. Haragus, A bifurcation theory for three-dimensional oblique travelling gravity-capillary water waves, J. Nonlinear Sci., 13 (2003), 397-447.doi: 10.1007/s00332-003-0530-8.


    M. D. Groves, M. Haragus and S. Sun, A dimension-breaking phenomenon in the theory of steady gravity-capillary water waves, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 360 (2002), 2189-2243.doi: 10.1098/rsta.2002.1066.


    M. D. Groves and A. Mielke, A spatial dynamics approach to three-dimensional gravity-capillary steady water waves, Proc. Roy. Soc. Edinburgh Sect., 131A (2001), 83-136.doi: 10.1017/S0308210500000809.


    M. D. Groves and S. M. Sun, Fully localised solitary-wave solutions of the three-dimensional gravity-capillary water-wave problem, Arch. Ration. Mech. Anal., 188 (2008), 1-91.doi: 10.1007/s00205-007-0085-1.


    M. Haragus and K. Kirchgässner, Three-Dimensional steady capillary-gravity waves, in Ergodic Theory, Analysis and Efficient Simulation of Dynamical Systems (ed. B. Fiedler), Springer-Verlag, Berlin, (2001), 363-397.


    G. Iooss and K. Kirchgässner, Bifurcation d'ondes solitaires en présence d'une faible tension superficielle, C. R. Acad. Sci. Paris Sér. I Math., 311 (1990), 265-268.


    G. Iooss and K. Kirchgässner, Water waves for small surface tension: an approach via normal form, Proc. Royal Soc. Edinburgh Sect., 122A (1992), 267-299.doi: 10.1017/S0308210500021119.


    G. Iooss and M. C. Pérouème, Perturbed homoclinic solutions in reversible 1:1 resonance vector fields, J. Diff. Equ., 102 (1993), 62-88.doi: 10.1006/jdeq.1993.1022.


    G. Iooss and P. Plotnikov, Asymmetrical three-dimensional travelling gravity waves, Arch. Ration. Mech. Anal., 200 (2011), 789-880.doi: 10.1007/s00205-010-0372-0.


    G. Keady and J. Norbury, On the existence theory for irrotational water waves, Math. Proc. Cambridge Philos. Soc., 83 (1978), 137-157.doi: 10.1017/S0305004100054372.


    Ju. P. Krasovskii, On the theory of steady-state waves of finite amplitude, Comp. Math. Math. Phys., 1 (1961), 836-855.


    E. Lombardi, Orbits homoclinic to exponentially small periodic orbits for a class of reversible systems. Application to water waves, Arch. Rat. Mech. Anal., 137 (1997), 227-304.doi: 10.1007/s002050050029.


    J. Reeder and M. Shinbrot, Three-dimensional, nonlinear wave interaction in water of constant depth, Nonlinear Anal., T.M.A., 5 (1981), 303-323.doi: 10.1016/0362-546X(81)90035-3.


    J. Shatah and C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler equation, Comm. Pure Appl. Math., 61 (2008), 698-744.doi: 10.1002/cpa.20213.


    S. Sun, Existence of a generalized solitary wave solution for water with positive Bond number smaller than 1/3, J. Math. Anal. Appl., 156 (1991), 471-504.doi: 10.1016/0022-247X(91)90410-2.


    S. Sun, Existence of large amplitude periodic waves in two-fluid flows of infinite depth, SIAM J. Math. Anal., 32 (2001), 1014-1031.doi: 10.1137/S0036141099352728.


    V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190-194.doi: 10.1007/BF00913182.

  • 加载中

Article Metrics

HTML views() PDF downloads(79) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint