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

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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

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

Received October 2012 Revised December 2013 Published March 2014

Abstract
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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 - A, 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1137/09077922X. Google Scholar
[11]	S. Deng, Two-dimensional travelling waves over a moving bottom,, J. Diff. Equa., 248 (2010), 1777. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/s00205-007-0085-1. Google Scholar
[20]	M. Haragus and K. Kirchgässner, Three-Dimensional steady capillary-gravity waves,, in Ergodic Theory, (2001), 363. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/s002050050029. Google Scholar
[28]	J. Reeder and M. Shinbrot, Three-dimensional, nonlinear wave interaction in water of constant depth,, Nonlinear Anal., 5 (1981), 303. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1137/09077922X. Google Scholar
[11]	S. Deng, Two-dimensional travelling waves over a moving bottom,, J. Diff. Equa., 248 (2010), 1777. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/s00205-007-0085-1. Google Scholar
[20]	M. Haragus and K. Kirchgässner, Three-Dimensional steady capillary-gravity waves,, in Ergodic Theory, (2001), 363. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/s002050050029. Google Scholar
[28]	J. Reeder and M. Shinbrot, Three-dimensional, nonlinear wave interaction in water of constant depth,, Nonlinear Anal., 5 (1981), 303. 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. 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. 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. 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. doi: 10.1007/BF00913182. Google Scholar

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