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September  2014, 34(9): 3419-3435. doi: 10.3934/dcds.2014.34.3419

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

Shengfu Deng 1,

1. 

Department of Mathematics, Zhanjiang Normal University, Zhanjiang, Guangdong 524048, China

Received  October 2012 Revised  December 2013 Published  March 2014

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.
Keywords: generalized Pitchfork bifurcation, Dirichlet-Neumann operator, periodic orbits, method of Lyapunov-Schmidt reduction..
Mathematics Subject Classification: Primary: 35B10, 35B32; Secondary: 76B1.
Citation: Shengfu Deng. Generalized pitchfork bifurcation on a two-dimensional gaseous star with self-gravity and surface tension. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3419-3435. doi: 10.3934/dcds.2014.34.3419
References:
[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.  Google Scholar

[2]

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.  Google Scholar

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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

[16]

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.  Google Scholar

[17]

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.  Google Scholar

[18]

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.  Google Scholar

[19]

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.  Google Scholar

[20]

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.  Google Scholar

[21]

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.  Google Scholar

[22]

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.  Google Scholar

[23]

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.  Google Scholar

[24]

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.  Google Scholar

[25]

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.  Google Scholar

[26]

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

[27]

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.  Google Scholar

[28]

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.  Google Scholar

[29]

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.  Google Scholar

[30]

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.  Google Scholar

[31]

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.  Google Scholar

[32]

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.  Google Scholar

show all references

References:
[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.  Google Scholar

[2]

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.  Google Scholar

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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

[16]

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.  Google Scholar

[17]

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.  Google Scholar

[18]

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.  Google Scholar

[19]

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.  Google Scholar

[20]

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.  Google Scholar

[21]

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.  Google Scholar

[22]

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.  Google Scholar

[23]

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.  Google Scholar

[24]

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.  Google Scholar

[25]

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.  Google Scholar

[26]

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

[27]

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.  Google Scholar

[28]

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.  Google Scholar

[29]

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.  Google Scholar

[30]

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.  Google Scholar

[31]

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.  Google Scholar

[32]

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.  Google Scholar

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