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November  2020, 13(11): 3189-3204. doi: 10.3934/dcdss.2020114

Existence of generalized homoclinic solutions for a modified Swift-Hohenberg equation

Yixia Shi 1, and  Maoan Han 2,1,,

1. 

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2. 

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China

* Corresponding author: Maoan Han

Received  September 2018 Revised  December 2018 Published  November 2020 Early access  October 2019

Fund Project: The corresponding author is supported by National Natural Science Foundation of China (11431008 and 11771296)

In this paper, we investigate the modified steady Swift-Hohenberg equation
$ \begin{equation} \begin{split} ku_{xxxx}+2ku_{xx}+\alpha u^{2}_{x}-\varepsilon u+u^{3} = 0,~~~~(1) \end{split} \end{equation} $
where
$ k>0 $
,
$ \alpha $
 and
$ \varepsilon $
 are constants. We obtain a homoclinic solution about the dominant system which will be proved to deform a reversible homoclinic solution approaching to a periodic solution of the whole equation with the aid of the Fourier series expansion method, the fixed point theorem, the reversibility and adjusting the phase shift. And the homoclinic solution approaching to a periodic solution of the equation are called generalized homoclinic solution.
Keywords: Modified Swift-Hohenberg system, generalized homoclinic solution, Fourier series expansion method, main system, reversibility.
Mathematics Subject Classification: Primary: 34C25, 37G15; Secondary: 37K50.
Citation: Yixia Shi, Maoan Han. Existence of generalized homoclinic solutions for a modified Swift-Hohenberg equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3189-3204. doi: 10.3934/dcdss.2020114
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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