• Previous Article
    On a generalized maximum principle for a transport-diffusion model with $\log$-modulated fractional dissipation
  • DCDS Home
  • This Issue
  • Next Article
    Justification of leading order quasicontinuum approximations of strongly nonlinear lattices
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

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.
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

[1]

Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201

[2]

Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739

[3]

Mei Ming. Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6039-6067. doi: 10.3934/dcds.2019264

[4]

Sándor Kelemen, Pavol Quittner. Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (3) : 731-740. doi: 10.3934/cpaa.2010.9.731

[5]

Mahamadi Warma. A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2043-2067. doi: 10.3934/cpaa.2015.14.2043

[6]

Kevin Arfi, Anna Rozanova-Pierrat. Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 1-26. doi: 10.3934/dcdss.2019001

[7]

Fengming Ma, Yiju Wang, Hongge Zhao. A potential reduction method for the generalized linear complementarity problem over a polyhedral cone. Journal of Industrial & Management Optimization, 2010, 6 (1) : 259-267. doi: 10.3934/jimo.2010.6.259

[8]

Dmitri Scheglov. Growth of periodic orbits and generalized diagonals for typical triangular billiards. Journal of Modern Dynamics, 2013, 7 (1) : 31-44. doi: 10.3934/jmd.2013.7.31

[9]

Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026

[10]

Anthony W. Baker, Michael Dellnitz, Oliver Junge. Topological method for rigorously computing periodic orbits using Fourier modes. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 901-920. doi: 10.3934/dcds.2005.13.901

[11]

Daniel Wilczak, Piotr Zgliczyński. Topological method for symmetric periodic orbits for maps with a reversing symmetry. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 629-652. doi: 10.3934/dcds.2007.17.629

[12]

P.E. Kloeden. Pitchfork and transcritical bifurcations in systems with homogeneous nonlinearities and an almost periodic time coefficient. Communications on Pure & Applied Analysis, 2004, 3 (2) : 161-173. doi: 10.3934/cpaa.2004.3.161

[13]

Zuowei Cai, Jianhua Huang, Lihong Huang. Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3591-3614. doi: 10.3934/dcdsb.2017181

[14]

Elvira Zappale. A note on dimension reduction for unbounded integrals with periodic microstructure via the unfolding method for slender domains. Evolution Equations & Control Theory, 2017, 6 (2) : 299-318. doi: 10.3934/eect.2017016

[15]

Fatima Ezzahra Lembarki, Jaume Llibre. Periodic orbits for a generalized Friedmann-Robertson-Walker Hamiltonian system in dimension $6$. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1165-1211. doi: 10.3934/dcdss.2015.8.1165

[16]

Robert Carlson. Dirichlet to Neumann maps for infinite quantum graphs. Networks & Heterogeneous Media, 2012, 7 (3) : 483-501. doi: 10.3934/nhm.2012.7.483

[17]

Shouchuan Hu, Nikolaos S. Papageorgiou. Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2889-2922. doi: 10.3934/cpaa.2013.12.2889

[18]

Weisheng Wu. Modified Schmidt games and non-dense forward orbits of partially hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3463-3481. doi: 10.3934/dcds.2016.36.3463

[19]

Shouchuan Hu, Nikolaos S. Papageorgiou. Nonlinear Neumann equations driven by a nonhomogeneous differential operator. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1055-1078. doi: 10.3934/cpaa.2011.10.1055

[20]

Juan Sánchez, Marta Net, José M. Vega. Amplitude equations close to a triple-(+1) bifurcation point of D4-symmetric periodic orbits in O(2)-equivariant systems. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1357-1380. doi: 10.3934/dcdsb.2006.6.1357

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]