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

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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

Abstract
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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 and 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.
[2]	M. Haragus and G. Iooss, "Local Bifurcations, Center Manifolds, and Normal Forms in Infinite Dimensional Dynamical Systems," Universitext, Springer-Verlag, EDP Sciences, 2011.
[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.
[4]	G. Iooss and M. Adelmeyer, "Topics in Bifurcation Theory and Applications," 2nd edition, Adv. Ser. World Sci., 1998.
[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.
[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.
[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.
[8]	E. Lombardi, "Oscillatory Integrals and Phenomena Beyond all Algebraic Orders," Lect. Notes in Math. 1741, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104102.
[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.
[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.
[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.
[12]	D. Ruelle and F. Takens, On the nature of turbulence, Com. Math. Phys., 20 (1971), 167-192. doi: 10.1007/BF01646553.
[13]	J. Swift and P. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev., A15 (1977), 319-328. doi: 10.1103/PhysRevA.15.319.
[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.
[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.
[16]	K. Yosida, "Fuctional Analysis," Reprint of the $6^{th}$ (1980) edition, Springer-Verlag, 1995.

show all references

References:

[1]	S. Agmon, "Lectures on Elliptic Boundary Value Problems," Princeton, N.J. Van Nostrand 1965.
[2]	M. Haragus and G. Iooss, "Local Bifurcations, Center Manifolds, and Normal Forms in Infinite Dimensional Dynamical Systems," Universitext, Springer-Verlag, EDP Sciences, 2011.
[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.
[4]	G. Iooss and M. Adelmeyer, "Topics in Bifurcation Theory and Applications," 2nd edition, Adv. Ser. World Sci., 1998.
[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.
[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.
[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.
[8]	E. Lombardi, "Oscillatory Integrals and Phenomena Beyond all Algebraic Orders," Lect. Notes in Math. 1741, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104102.
[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.
[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.
[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.
[12]	D. Ruelle and F. Takens, On the nature of turbulence, Com. Math. Phys., 20 (1971), 167-192. doi: 10.1007/BF01646553.
[13]	J. Swift and P. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev., A15 (1977), 319-328. doi: 10.1103/PhysRevA.15.319.
[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.
[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.
[16]	K. Yosida, "Fuctional Analysis," Reprint of the $6^{th}$ (1980) edition, Springer-Verlag, 1995.

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