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Existence of generalized homoclinic solutions for a modified Swift-Hohenberg equation
1. | Department of Mathematics, Shanghai Normal University, Shanghai 200234, China |
2. | Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China |
$ \begin{equation} \begin{split} ku_{xxxx}+2ku_{xx}+\alpha u^{2}_{x}-\varepsilon u+u^{3} = 0,~~~~(1) \end{split} \end{equation} $ |
$ k>0 $ |
$ \alpha $ |
$ \varepsilon $ |
References:
[1] |
E. Bodenschatz, W. Pesch and G. Ahlers,
Recent developments in Rayleigh-Bénard convection, Annu. Rev. Fluid Mech., Annual Reviews, Palo Alto, CA, 32 (2000), 709-778.
doi: 10.1146/annurev.fluid.32.1.709. |
[2] |
F. H. Busse,
Non-linear properties of thermal convection, Rep. Progr. Phys., 41 (1978), 1929-1967.
doi: 10.1088/0034-4885/41/12/003. |
[3] |
S. F. Deng, B. L. Guo and X. P. Li,
Notes on homoclinic solutions of the steady Swift-Hohenberg equation, Chin. Ann. Math. Ser. B, 34 (2013), 917-920.
doi: 10.1007/s11401-013-0801-0. |
[4] |
S. F. Deng,
Periodic solutions and homoclinic solutions for a Swift-Hohenberg equation with dispersion, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1647-1662.
doi: 10.3934/dcdss.2016068. |
[5] |
S. F. Deng and B. L. Guo,
Generalized homoclinic solutions of a coupled Schrödinger system under a small perturbation, J. Dyn. Diff. Equat., 24 (2012), 761-776.
doi: 10.1007/s10884-012-9274-1. |
[6] |
S. F. Deng and X. P. Li,
Generalized homoclinic solutions for the Swift-Hohenberg equation, J. Math. Anal. Appl., 390 (2012), 15-26.
doi: 10.1016/j.jmaa.2011.11.074. |
[7] |
S. F. Deng and S.-M. 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. |
[8] |
S. F. Deng and S.-M. Sun,
Exact theory of three-dimensional water waves at the critical speed, SIAM J. Math. Anal., 42 (2010), 2721-2761.
doi: 10.1137/09077922X. |
[9] |
S. F. Deng, B. L. Guo and T. C. Wang,
Existence of generalized heteroclinic solutions of the coupled Schrödinger system under a small perturbation, Chin. Ann. Math. Ser. B, 35 (2014), 857-872.
doi: 10.1007/s11401-014-0867-3. |
[10] |
A. Doelman, B. Standstede, A. Scheel and G. Schneider,
Propagation of hexagonal patterns near onset, Eur. J. Appl. Math., 14 (2003), 85-110.
doi: 10.1017/S095679250200503X. |
[11] |
E. Lombardi, Oscillatory Integrals and Phenomena Beyond all Algebraic Orders. With Applications to Homoclinic Orbits in Reversible Systems, Lecture Notes in Mathematics, 1741. Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0104102. |
[12] |
E. Lombardi,
Orbits homoclinic to exponentially small periodic orbits for a class of reversible systems. Application to water waves, Arch. Rational Mech. Anal., 137 (1997), 227-304.
doi: 10.1007/s002050050029. |
[13] |
E. Lombardi,
Homoclinic orbits to small periodic orbits for a class of reversible systems, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 1035-1054.
doi: 10.1017/S0308210500023246. |
[14] |
D. Y. Hsieh, S. Tang and X. Wang,
On hydrodynamic instabilities, chaos and phase transition, Acta Mech. Sinica, 12 (1996), 1-14.
|
[15] |
D. Y. Hsieh,
Elemental mechanisms of hydrodynamic instabilities, Acta Mech. Sinica, 10 (1994), 193-202.
doi: 10.1007/BF02487607. |
[16] |
H. Kielhöfer, Bifurcation Theory: An Introduction With Applications to PDEs, Springer, New York, 2003. |
[17] |
Y. Kuramoto,
Diffusion-induced chaos in reaction systems, Supp. Prog. Theoret. Phys., 64 (1978), 346-367.
doi: 10.1143/PTPS.64.346. |
[18] |
R. E. La Quey, P. H. Rutherford and W. M. Tang,
Nonlinear saturation of the trapped-ton mode, Phys. Rev. Lett., 34 (1975), 391-394.
|
[19] |
J. Lega, J. V. Moloney and A. C. Newell,
Swift-Hohenberg equation for lasers, Phys. Rev. Lett., 73 (1994), 2978-2981.
doi: 10.1103/PhysRevLett.73.2978. |
[20] |
A. Mielke and G. Schneider,
Attractors for modulation equation on unbounded domains-existence and comparison, Nonlinearity, 8 (1995), 743-768.
doi: 10.1088/0951-7715/8/5/006. |
[21] |
M. Polat,
Global attractor for a modified Swift-Hohenberg equation, Comput. Math. Appl., 57 (2009), 62-66.
doi: 10.1016/j.camwa.2008.09.028. |
[22] |
Q. Ouyang and H. L. Swinney,
Onset and beyond turing pattern formation, Chemical Waves and Patterns, 10 (1995), 269-295.
doi: 10.1007/978-94-011-1156-0_8. |
[23] |
S. H. Park and J. Y. Park,
Pullback attractor for a non-autonomous modified Swift-Hohenberg equation, Comput. Math. Appl., 67 (2014), 542-548.
doi: 10.1016/j.camwa.2013.11.011. |
[24] |
Y. X. Shi and S. F. Deng,
Existence of generalized homoclinic solutions of a coupled KdV-type Boussinesq system under a small perturbation, J. Apple. Anal. Comput., 7 (2017), 392-410.
|
[25] |
T. Shlang and G. L. Sivashinsky,
Irregular flow of a liquid film down a vertical column, J. Phys. France, 43 (1982), 459-466.
|
[26] |
G. I. Siivashinsky,
Nonlinear analysis of hydrodynamic instability in laminar flames. I. Derivation of basic equations, Acta Astron., 4 (1977), 1177-1206.
doi: 10.1016/0094-5765(77)90096-0. |
[27] |
L. Y. Song, Y. D. Zhang and T. Ma,
Global attractor of a modified Swift-Hohenberg equation in $H^k$ spaces, Non. Anal., 72 (2010), 183-191.
doi: 10.1016/j.na.2009.06.103. |
[28] |
A. M. Soward,
Bifurcation and stability of finite amplitude convection in a rotating layer, Phys. D., 14 (1985), 227-241.
doi: 10.1016/0167-2789(85)90181-2. |
[29] |
J. Swift and P. C. Hohenberg,
Hydrodynamics fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319-328.
doi: 10.1103/PhysRevA.15.319. |
[30] |
W. Walter, Gewöhnliche Differentialgleichungen. Eine Einführung, Heidelberger Taschenbücher, Band 110. Springer-Verlag, Berlin-New York, 1972. |
[31] |
Q. K. Xiao and H. J. Gao,
Bifurcation analysis of a modified Swift-Hohenberg equation, Nonlinear Anal. Real World Appl., 11 (2010), 4451-4464.
doi: 10.1016/j.nonrwa.2010.05.028. |
[32] |
L. Xu and Q. Z. Ma, Existence of the uniform attractors for a non-autonomous modified Swift-Hohenberg equation, Adv. in Difference Equations, 2015 (2015), 1-11.
doi: 10.1186/s13662-015-0492-9. |
[33] |
W. B. Zhang and J. Viñals,
Pattern formation in weakly damped parametric surface waves, J. Fluid Mech., 336 (1997), 301-330.
doi: 10.1017/S0022112096004764. |
[34] |
X. P. Zhao, B. Liu, P. Zhang, W. Y. Zhang and F. N. Liu, Fourier spectral method for the modified Swift-Hohenberg equation, Adv. Difference Equ., 2013 (2013), 19 pp.
doi: 10.1186/1687-1847-2013-156. |
show all references
References:
[1] |
E. Bodenschatz, W. Pesch and G. Ahlers,
Recent developments in Rayleigh-Bénard convection, Annu. Rev. Fluid Mech., Annual Reviews, Palo Alto, CA, 32 (2000), 709-778.
doi: 10.1146/annurev.fluid.32.1.709. |
[2] |
F. H. Busse,
Non-linear properties of thermal convection, Rep. Progr. Phys., 41 (1978), 1929-1967.
doi: 10.1088/0034-4885/41/12/003. |
[3] |
S. F. Deng, B. L. Guo and X. P. Li,
Notes on homoclinic solutions of the steady Swift-Hohenberg equation, Chin. Ann. Math. Ser. B, 34 (2013), 917-920.
doi: 10.1007/s11401-013-0801-0. |
[4] |
S. F. Deng,
Periodic solutions and homoclinic solutions for a Swift-Hohenberg equation with dispersion, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1647-1662.
doi: 10.3934/dcdss.2016068. |
[5] |
S. F. Deng and B. L. Guo,
Generalized homoclinic solutions of a coupled Schrödinger system under a small perturbation, J. Dyn. Diff. Equat., 24 (2012), 761-776.
doi: 10.1007/s10884-012-9274-1. |
[6] |
S. F. Deng and X. P. Li,
Generalized homoclinic solutions for the Swift-Hohenberg equation, J. Math. Anal. Appl., 390 (2012), 15-26.
doi: 10.1016/j.jmaa.2011.11.074. |
[7] |
S. F. Deng and S.-M. 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. |
[8] |
S. F. Deng and S.-M. Sun,
Exact theory of three-dimensional water waves at the critical speed, SIAM J. Math. Anal., 42 (2010), 2721-2761.
doi: 10.1137/09077922X. |
[9] |
S. F. Deng, B. L. Guo and T. C. Wang,
Existence of generalized heteroclinic solutions of the coupled Schrödinger system under a small perturbation, Chin. Ann. Math. Ser. B, 35 (2014), 857-872.
doi: 10.1007/s11401-014-0867-3. |
[10] |
A. Doelman, B. Standstede, A. Scheel and G. Schneider,
Propagation of hexagonal patterns near onset, Eur. J. Appl. Math., 14 (2003), 85-110.
doi: 10.1017/S095679250200503X. |
[11] |
E. Lombardi, Oscillatory Integrals and Phenomena Beyond all Algebraic Orders. With Applications to Homoclinic Orbits in Reversible Systems, Lecture Notes in Mathematics, 1741. Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0104102. |
[12] |
E. Lombardi,
Orbits homoclinic to exponentially small periodic orbits for a class of reversible systems. Application to water waves, Arch. Rational Mech. Anal., 137 (1997), 227-304.
doi: 10.1007/s002050050029. |
[13] |
E. Lombardi,
Homoclinic orbits to small periodic orbits for a class of reversible systems, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 1035-1054.
doi: 10.1017/S0308210500023246. |
[14] |
D. Y. Hsieh, S. Tang and X. Wang,
On hydrodynamic instabilities, chaos and phase transition, Acta Mech. Sinica, 12 (1996), 1-14.
|
[15] |
D. Y. Hsieh,
Elemental mechanisms of hydrodynamic instabilities, Acta Mech. Sinica, 10 (1994), 193-202.
doi: 10.1007/BF02487607. |
[16] |
H. Kielhöfer, Bifurcation Theory: An Introduction With Applications to PDEs, Springer, New York, 2003. |
[17] |
Y. Kuramoto,
Diffusion-induced chaos in reaction systems, Supp. Prog. Theoret. Phys., 64 (1978), 346-367.
doi: 10.1143/PTPS.64.346. |
[18] |
R. E. La Quey, P. H. Rutherford and W. M. Tang,
Nonlinear saturation of the trapped-ton mode, Phys. Rev. Lett., 34 (1975), 391-394.
|
[19] |
J. Lega, J. V. Moloney and A. C. Newell,
Swift-Hohenberg equation for lasers, Phys. Rev. Lett., 73 (1994), 2978-2981.
doi: 10.1103/PhysRevLett.73.2978. |
[20] |
A. Mielke and G. Schneider,
Attractors for modulation equation on unbounded domains-existence and comparison, Nonlinearity, 8 (1995), 743-768.
doi: 10.1088/0951-7715/8/5/006. |
[21] |
M. Polat,
Global attractor for a modified Swift-Hohenberg equation, Comput. Math. Appl., 57 (2009), 62-66.
doi: 10.1016/j.camwa.2008.09.028. |
[22] |
Q. Ouyang and H. L. Swinney,
Onset and beyond turing pattern formation, Chemical Waves and Patterns, 10 (1995), 269-295.
doi: 10.1007/978-94-011-1156-0_8. |
[23] |
S. H. Park and J. Y. Park,
Pullback attractor for a non-autonomous modified Swift-Hohenberg equation, Comput. Math. Appl., 67 (2014), 542-548.
doi: 10.1016/j.camwa.2013.11.011. |
[24] |
Y. X. Shi and S. F. Deng,
Existence of generalized homoclinic solutions of a coupled KdV-type Boussinesq system under a small perturbation, J. Apple. Anal. Comput., 7 (2017), 392-410.
|
[25] |
T. Shlang and G. L. Sivashinsky,
Irregular flow of a liquid film down a vertical column, J. Phys. France, 43 (1982), 459-466.
|
[26] |
G. I. Siivashinsky,
Nonlinear analysis of hydrodynamic instability in laminar flames. I. Derivation of basic equations, Acta Astron., 4 (1977), 1177-1206.
doi: 10.1016/0094-5765(77)90096-0. |
[27] |
L. Y. Song, Y. D. Zhang and T. Ma,
Global attractor of a modified Swift-Hohenberg equation in $H^k$ spaces, Non. Anal., 72 (2010), 183-191.
doi: 10.1016/j.na.2009.06.103. |
[28] |
A. M. Soward,
Bifurcation and stability of finite amplitude convection in a rotating layer, Phys. D., 14 (1985), 227-241.
doi: 10.1016/0167-2789(85)90181-2. |
[29] |
J. Swift and P. C. Hohenberg,
Hydrodynamics fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319-328.
doi: 10.1103/PhysRevA.15.319. |
[30] |
W. Walter, Gewöhnliche Differentialgleichungen. Eine Einführung, Heidelberger Taschenbücher, Band 110. Springer-Verlag, Berlin-New York, 1972. |
[31] |
Q. K. Xiao and H. J. Gao,
Bifurcation analysis of a modified Swift-Hohenberg equation, Nonlinear Anal. Real World Appl., 11 (2010), 4451-4464.
doi: 10.1016/j.nonrwa.2010.05.028. |
[32] |
L. Xu and Q. Z. Ma, Existence of the uniform attractors for a non-autonomous modified Swift-Hohenberg equation, Adv. in Difference Equations, 2015 (2015), 1-11.
doi: 10.1186/s13662-015-0492-9. |
[33] |
W. B. Zhang and J. Viñals,
Pattern formation in weakly damped parametric surface waves, J. Fluid Mech., 336 (1997), 301-330.
doi: 10.1017/S0022112096004764. |
[34] |
X. P. Zhao, B. Liu, P. Zhang, W. Y. Zhang and F. N. Liu, Fourier spectral method for the modified Swift-Hohenberg equation, Adv. Difference Equ., 2013 (2013), 19 pp.
doi: 10.1186/1687-1847-2013-156. |
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