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Remarks on a two dimensional BBM type equation
The dynamics of small amplitude solutions of the Swift-Hohenberg equation on a large interval
1. | College of Science, Wuhan University of Science and Technology, Wuhan 430065, China |
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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