American Institute of Mathematical Sciences

Advanced
  • 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

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 - 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

Tools

Metrics

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

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences