December  2016, 9(6): 1647-1662. doi: 10.3934/dcdss.2016068

Periodic solutions and homoclinic solutions for a Swift-Hohenberg equation with dispersion

1. 

Department of Mathematics, Zhanjiang Normal University, Zhanjiang, Guangdong 524048

Received  July 2015 Revised  August 2016 Published  November 2016

We investigate the 1D Swift-Hohenberg equation with dispersion $$u_t+2u_{\xi\xi}-\sigma u_{\xi\xi\xi}+u_{\xi\xi\xi\xi}=\alpha u+\beta u^2-\gamma u^3,$$ where $\sigma, \alpha, \beta$ and $\gamma$ are constants. Even if only the stationary solutions of this equation are considered, the dispersion term $-\sigma u_{\xi\xi\xi}$ destroys the spatial reversibility which plays an important role for studying localized patterns. In this paper, we focus on its traveling wave solutions and directly apply the dynamical approach to provide the first rigorous proof of existence of the periodic solutions and the homoclinic solutions bifurcating from the origin without the reversibility condition as the parameters are varied.
Citation: Shengfu Deng. Periodic solutions and homoclinic solutions for a Swift-Hohenberg equation with dispersion. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1647-1662. doi: 10.3934/dcdss.2016068
References:
[1]

D. Avitabile, D. J. B. Lloyd, J. Burke, E. Knobloch and B. Sandstede, To snake or not to snake in the planar Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 9 (2010), 704-733. doi: 10.1137/100782747.

[2]

M. Beck, J. Knobloch, D. J. B. Lloyd, B. Sandstede and T. Wagenknecht, Snakes, ladders, and isolas of localized patterns, SIAM J. Math. Anal., 41 (2009), 936-972. doi: 10.1137/080713306.

[3]

D. Blair, I. S. Aranson, G. W. Crabtree, V. Vinokur, L. S. Tsimring and C. Josserand, Patterns in thin vibrated granular layers: Interfaces, hexagons, and superoscillons, Phys. Rev. E, 61 (2000), 5600-5610. doi: 10.1103/PhysRevE.61.5600.

[4]

B. Braaksma, G. Iooss and L. Stolovitch, Existence of quasipattern solutions of the Swift-Hohenberg equation, Arch. Ration. Mech. Anal., 209 (2013), 255-285. doi: 10.1007/s00205-013-0627-7.

[5]

J. Burke, S. M. Houghton and E. Knobloch, Swift-Hohenberg equation with broken reflection symmetry, Phys. Rev. E, 80 (2009), 036202. doi: 10.1103/PhysRevE.80.036202.

[6]

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals For Engineers and Physicists, Springer-Verlag, Berlin, 1954.

[7]

P. Collet and J. P. Eckmann, Instabilities and Fronts in Extended Systems, Princeton University Press, Princeton, 1990. doi: 10.1515/9781400861026.

[8]

S. Day, Y. Hiraoka, K. Mischaikow and T. Ogawa, Rigorous numerics for global dynamics: A study of the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 4 (2005), 1-31. doi: 10.1137/040604479.

[9]

S. Deng and X. 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.

[10]

S. Deng and S. M. Sun, Multi-hump solutions with small oscillations at infinity for stationary Swift-Hohenberg equation, submitted.

[11]

J. P. Gaivão and V. Gelfreich, Splitting of separatrices for the Hamiltonian-Hopf bifurcation with the Swift-Hohenberg equation as an example, Nonlinearity, 24 (2011), 677-698. doi: 10.1088/0951-7715/24/3/002.

[12]

P. Gandhi, C. Beaume and E. Knobloch, A new resonance mechanism in the Swift-Hohenberg rquation with time-periodic forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 860-892. doi: 10.1137/14099468X.

[13]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1990.

[14]

M. Haragus and A. Scheel, Interfaces between rolls in the Swift-Hohenberg equation, Int. J. Dyn. Syst. Diff. Equ., 1 (2007), 89-97. doi: 10.1504/IJDSDE.2007.016510.

[15]

G. Iooss and A. M. Rucklidge, On the existence of quasipattern solutions of the Swift-Hohenberg equation, J. Nonlinear Sci., 20 (2010), 361-394. doi: 10.1007/s00332-010-9063-0.

[16]

J. Knobloch, M. Vielitz and T. Wagenknecht, Non-reversible perturbations of homoclinic snaking scenarios, Nonlinearity, 25 (2012), 3469-3485. doi: 10.1088/0951-7715/25/12/3469.

[17]

N. A. Kudryashov and D. I. Sinelshchikov, Exact solutions of the Swift-Hohenberg equation with dispersion, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 26-34. doi: 10.1016/j.cnsns.2011.04.008.

[18]

R. E. LaQuey, S. M. Mahajan, P. H. Rutherford and W. M. Tang, Nonlinear saturation of the trapped-ion mode, Phys. Rev. Lett., 34 (1975), 391-394. doi: 10.1103/PhysRevLett.34.391.

[19]

L. Lee and H. Swinney, Lamellar structures and self-replicating spots in a reaction-diffusion system, Phys. Rev. E, 51 (1995), 1899-1915. doi: 10.1103/PhysRevE.51.1899.

[20]

L. 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.

[21]

M. Lopez-Fernandez and S. Sauter, Fast and stable contour integration for high order divided differences via elliptic functions, Math. Comp., 84 (2015), 1291-1315. doi: 10.1090/S0025-5718-2014-02890-1.

[22]

E. Makrides and B. Sandstede, Predicting the bifurcation structure of localized snaking patterns, Phys. D, 268 (2014), 59-78. doi: 10.1016/j.physd.2013.11.009.

[23]

P. Mandel, Theoretical Problems in Cavity Nonlinear Optics, Cambridge University Press, Cambridge, 1997.

[24]

S. G. McCalla and B. Sandstede, Spots in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 12 (2013), 831-877. doi: 10.1137/120882111.

[25]

A. Mielke, Instability and stability of rolls in the Swift-Hohenberg equation, Comm. Math. Phys., 189 (1997), 829-853. doi: 10.1007/s002200050230.

[26]

D. Morgan and J. H. P. Dawes, The Swift-Hohenberg equation with a nonlocal nonlinearity, Phys. D, 270 (2014), 60-80. doi: 10.1016/j.physd.2013.11.018.

[27]

L. A. Peletier and V. Rottschafer, Pattern selection of solutions of the Swift-Hohenberg equation, Phys. D, 194 (2004), 95-126. doi: 10.1016/j.physd.2004.01.043.

[28]

L. A. Peletier and J. F. Williams, Some canonical bifurcations in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 6 (2007), 208-235. doi: 10.1137/050647232.

[29]

D. Smets and J. B. van den Berg, Homoclinic solutions for Swift-Hohenberg and suspension bridge type equations, J. Diff. Eqns., 184 (2002), 78-96. doi: 10.1006/jdeq.2001.4135.

[30]

J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319-328. doi: 10.1103/PhysRevA.15.319.

[31]

J. B. van den Berg, L. A. Peletier and W. C. Troy, Global branches of multi-bump periodic solutions of the Swift-Hohenberg equation, Arch. Ration. Mech. Anal., 158 (2001), 91-153. doi: 10.1007/PL00004243.

[32]

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, Universitext, 1996. doi: 10.1007/978-3-642-61453-8.

show all references

References:
[1]

D. Avitabile, D. J. B. Lloyd, J. Burke, E. Knobloch and B. Sandstede, To snake or not to snake in the planar Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 9 (2010), 704-733. doi: 10.1137/100782747.

[2]

M. Beck, J. Knobloch, D. J. B. Lloyd, B. Sandstede and T. Wagenknecht, Snakes, ladders, and isolas of localized patterns, SIAM J. Math. Anal., 41 (2009), 936-972. doi: 10.1137/080713306.

[3]

D. Blair, I. S. Aranson, G. W. Crabtree, V. Vinokur, L. S. Tsimring and C. Josserand, Patterns in thin vibrated granular layers: Interfaces, hexagons, and superoscillons, Phys. Rev. E, 61 (2000), 5600-5610. doi: 10.1103/PhysRevE.61.5600.

[4]

B. Braaksma, G. Iooss and L. Stolovitch, Existence of quasipattern solutions of the Swift-Hohenberg equation, Arch. Ration. Mech. Anal., 209 (2013), 255-285. doi: 10.1007/s00205-013-0627-7.

[5]

J. Burke, S. M. Houghton and E. Knobloch, Swift-Hohenberg equation with broken reflection symmetry, Phys. Rev. E, 80 (2009), 036202. doi: 10.1103/PhysRevE.80.036202.

[6]

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals For Engineers and Physicists, Springer-Verlag, Berlin, 1954.

[7]

P. Collet and J. P. Eckmann, Instabilities and Fronts in Extended Systems, Princeton University Press, Princeton, 1990. doi: 10.1515/9781400861026.

[8]

S. Day, Y. Hiraoka, K. Mischaikow and T. Ogawa, Rigorous numerics for global dynamics: A study of the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 4 (2005), 1-31. doi: 10.1137/040604479.

[9]

S. Deng and X. 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.

[10]

S. Deng and S. M. Sun, Multi-hump solutions with small oscillations at infinity for stationary Swift-Hohenberg equation, submitted.

[11]

J. P. Gaivão and V. Gelfreich, Splitting of separatrices for the Hamiltonian-Hopf bifurcation with the Swift-Hohenberg equation as an example, Nonlinearity, 24 (2011), 677-698. doi: 10.1088/0951-7715/24/3/002.

[12]

P. Gandhi, C. Beaume and E. Knobloch, A new resonance mechanism in the Swift-Hohenberg rquation with time-periodic forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 860-892. doi: 10.1137/14099468X.

[13]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1990.

[14]

M. Haragus and A. Scheel, Interfaces between rolls in the Swift-Hohenberg equation, Int. J. Dyn. Syst. Diff. Equ., 1 (2007), 89-97. doi: 10.1504/IJDSDE.2007.016510.

[15]

G. Iooss and A. M. Rucklidge, On the existence of quasipattern solutions of the Swift-Hohenberg equation, J. Nonlinear Sci., 20 (2010), 361-394. doi: 10.1007/s00332-010-9063-0.

[16]

J. Knobloch, M. Vielitz and T. Wagenknecht, Non-reversible perturbations of homoclinic snaking scenarios, Nonlinearity, 25 (2012), 3469-3485. doi: 10.1088/0951-7715/25/12/3469.

[17]

N. A. Kudryashov and D. I. Sinelshchikov, Exact solutions of the Swift-Hohenberg equation with dispersion, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 26-34. doi: 10.1016/j.cnsns.2011.04.008.

[18]

R. E. LaQuey, S. M. Mahajan, P. H. Rutherford and W. M. Tang, Nonlinear saturation of the trapped-ion mode, Phys. Rev. Lett., 34 (1975), 391-394. doi: 10.1103/PhysRevLett.34.391.

[19]

L. Lee and H. Swinney, Lamellar structures and self-replicating spots in a reaction-diffusion system, Phys. Rev. E, 51 (1995), 1899-1915. doi: 10.1103/PhysRevE.51.1899.

[20]

L. 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.

[21]

M. Lopez-Fernandez and S. Sauter, Fast and stable contour integration for high order divided differences via elliptic functions, Math. Comp., 84 (2015), 1291-1315. doi: 10.1090/S0025-5718-2014-02890-1.

[22]

E. Makrides and B. Sandstede, Predicting the bifurcation structure of localized snaking patterns, Phys. D, 268 (2014), 59-78. doi: 10.1016/j.physd.2013.11.009.

[23]

P. Mandel, Theoretical Problems in Cavity Nonlinear Optics, Cambridge University Press, Cambridge, 1997.

[24]

S. G. McCalla and B. Sandstede, Spots in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 12 (2013), 831-877. doi: 10.1137/120882111.

[25]

A. Mielke, Instability and stability of rolls in the Swift-Hohenberg equation, Comm. Math. Phys., 189 (1997), 829-853. doi: 10.1007/s002200050230.

[26]

D. Morgan and J. H. P. Dawes, The Swift-Hohenberg equation with a nonlocal nonlinearity, Phys. D, 270 (2014), 60-80. doi: 10.1016/j.physd.2013.11.018.

[27]

L. A. Peletier and V. Rottschafer, Pattern selection of solutions of the Swift-Hohenberg equation, Phys. D, 194 (2004), 95-126. doi: 10.1016/j.physd.2004.01.043.

[28]

L. A. Peletier and J. F. Williams, Some canonical bifurcations in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 6 (2007), 208-235. doi: 10.1137/050647232.

[29]

D. Smets and J. B. van den Berg, Homoclinic solutions for Swift-Hohenberg and suspension bridge type equations, J. Diff. Eqns., 184 (2002), 78-96. doi: 10.1006/jdeq.2001.4135.

[30]

J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319-328. doi: 10.1103/PhysRevA.15.319.

[31]

J. B. van den Berg, L. A. Peletier and W. C. Troy, Global branches of multi-bump periodic solutions of the Swift-Hohenberg equation, Arch. Ration. Mech. Anal., 158 (2001), 91-153. doi: 10.1007/PL00004243.

[32]

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, Universitext, 1996. doi: 10.1007/978-3-642-61453-8.

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