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May  2012, 11(3): 1129-1156. doi: 10.3934/cpaa.2012.11.1129

The dynamics of small amplitude solutions of the Swift-Hohenberg equation on a large interval

Ling-Jun Wang 1,

1. 

College of Science, Wuhan University of Science and Technology, Wuhan 430065, China

Received  November 2010 Revised  March 2011 Published  December 2011

We study the dynamics of the one dimensional Swift-Hohenberg equation defined on a large interval $(-l,l)$ with Dirichlet-Neumann boundary conditions, where $l>0$ is large and lies outside of some small neighborhoods of the points $n\pi$ and $(n+1/2)\pi,n \in N$. The arguments are based on dynamical system formulation and bifurcation theory. We show that the system with Dirichlet-Neumann boundary conditions can be reduced to a two-dimensional center manifold for each bifurcation parameter $O(l^{-2})$-close to its critical values when $l$ is sufficiently large. On this invariant manifold, we find families of steady solutions and heteroclinic connections with each connecting two different steady solutions. Moreover, by comparing the above dynamics with that of the Swift-Hohenberg equation defined on $R$ and admitting $2\pi$-spatially periodic solutions in [4], we find that the dynamics in our case preserves the main features of the dynamics in the $2\pi$ spatially periodic case.
Keywords: Dirichlet-Neumann boundary., center manifold theory, Swift-Hohenberg equation, dynamics.
Mathematics Subject Classification: Primary: 37L10; Secondary: 35B3.
Citation: Ling-Jun Wang. The dynamics of small amplitude solutions of the Swift-Hohenberg equation on a large interval. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1129-1156. doi: 10.3934/cpaa.2012.11.1129
References:
[1]

S. Agmon, "Lectures on Elliptic Boundary Value Problems," Princeton, N.J. Van Nostrand 1965.  Google Scholar

[2]

M. Haragus and G. Iooss, "Local Bifurcations, Center Manifolds, and Normal Forms in Infinite Dimensional Dynamical Systems," Universitext, Springer-Verlag, EDP Sciences, 2011.  Google Scholar

[3]

P. C. Hohenberg and J. B. Swift, Effects of additive noise at the onset of Rayleigh-Bénard convection, Phys. Rev., A46 (1992), 4773-4785. Google Scholar

[4]

G. Iooss and M. Adelmeyer, "Topics in Bifurcation Theory and Applications," 2nd edition, Adv. Ser. World Sci., 1998.  Google Scholar

[5]

K. Kirchgässner, Wave solutions of reversible systems and applications, J. Diff. Eqns., 45 (1982), 113-127. doi: 10.1016/0022-0396(82)90058-4.  Google Scholar

[6]

O. Lanford III, Bifurcation of periodic solutions into invariant tori, in "Lect. Notes in Math.," 322, Springer-Verlag, (1973), 159-192. doi: 10.1007/BFb0060566.  Google Scholar

[7]

J. Lega, A. C. Newell and J. V. Moloney, Swift-hohenberg equation for lasers, Phys. Rev. Lett., 73 (1994), 2978-2981. doi: 10.1103/PhysRevLett.73.2978.  Google Scholar

[8]

E. Lombardi, "Oscillatory Integrals and Phenomena Beyond all Algebraic Orders," Lect. Notes in Math. 1741, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104102.  Google Scholar

[9]

A. Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci., 10 (1988), 51-66. doi: 10.1002/mma.1670100105.  Google Scholar

[10]

Y. Pomeau and P. Manneville, Wavelength selection in cellular flows, Phys. Letters., 75A (1980), 296-298. doi: 10.1016/0375-9601(80)90568-X.  Google Scholar

[11]

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.  Google Scholar

[12]

D. Ruelle and F. Takens, On the nature of turbulence, Com. Math. Phys., 20 (1971), 167-192. doi: 10.1007/BF01646553.  Google Scholar

[13]

J. Swift and P. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev., A15 (1977), 319-328. doi: 10.1103/PhysRevA.15.319.  Google Scholar

[14]

A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, Dynamics Reported: Expositions in Dynamical Systems, 1 (1992), 125-163. doi: 10.1007/978-3-642-61243-5_4.  Google Scholar

[15]

L.-J. Wang, Homoclinic and heteroclinic orbits for the $0^2$ or $0^2i \omega$ singularity in the presence of two reversibility symmetries, Quart. Appl. Math., 67 (2009), 1-38.  Google Scholar

[16]

K. Yosida, "Fuctional Analysis," Reprint of the $6^{th}$ (1980) edition, Springer-Verlag, 1995.  Google Scholar

show all references

References:
[1]

S. Agmon, "Lectures on Elliptic Boundary Value Problems," Princeton, N.J. Van Nostrand 1965.  Google Scholar

[2]

M. Haragus and G. Iooss, "Local Bifurcations, Center Manifolds, and Normal Forms in Infinite Dimensional Dynamical Systems," Universitext, Springer-Verlag, EDP Sciences, 2011.  Google Scholar

[3]

P. C. Hohenberg and J. B. Swift, Effects of additive noise at the onset of Rayleigh-Bénard convection, Phys. Rev., A46 (1992), 4773-4785. Google Scholar

[4]

G. Iooss and M. Adelmeyer, "Topics in Bifurcation Theory and Applications," 2nd edition, Adv. Ser. World Sci., 1998.  Google Scholar

[5]

K. Kirchgässner, Wave solutions of reversible systems and applications, J. Diff. Eqns., 45 (1982), 113-127. doi: 10.1016/0022-0396(82)90058-4.  Google Scholar

[6]

O. Lanford III, Bifurcation of periodic solutions into invariant tori, in "Lect. Notes in Math.," 322, Springer-Verlag, (1973), 159-192. doi: 10.1007/BFb0060566.  Google Scholar

[7]

J. Lega, A. C. Newell and J. V. Moloney, Swift-hohenberg equation for lasers, Phys. Rev. Lett., 73 (1994), 2978-2981. doi: 10.1103/PhysRevLett.73.2978.  Google Scholar

[8]

E. Lombardi, "Oscillatory Integrals and Phenomena Beyond all Algebraic Orders," Lect. Notes in Math. 1741, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104102.  Google Scholar

[9]

A. Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci., 10 (1988), 51-66. doi: 10.1002/mma.1670100105.  Google Scholar

[10]

Y. Pomeau and P. Manneville, Wavelength selection in cellular flows, Phys. Letters., 75A (1980), 296-298. doi: 10.1016/0375-9601(80)90568-X.  Google Scholar

[11]

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.  Google Scholar

[12]

D. Ruelle and F. Takens, On the nature of turbulence, Com. Math. Phys., 20 (1971), 167-192. doi: 10.1007/BF01646553.  Google Scholar

[13]

J. Swift and P. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev., A15 (1977), 319-328. doi: 10.1103/PhysRevA.15.319.  Google Scholar

[14]

A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, Dynamics Reported: Expositions in Dynamical Systems, 1 (1992), 125-163. doi: 10.1007/978-3-642-61243-5_4.  Google Scholar

[15]

L.-J. Wang, Homoclinic and heteroclinic orbits for the $0^2$ or $0^2i \omega$ singularity in the presence of two reversibility symmetries, Quart. Appl. Math., 67 (2009), 1-38.  Google Scholar

[16]

K. Yosida, "Fuctional Analysis," Reprint of the $6^{th}$ (1980) edition, Springer-Verlag, 1995.  Google Scholar

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