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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

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