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Bifurcation from two equilibria of steady state solutions for non-reversible amplitude equations. (English) Zbl 1450.35052

Summary: In this paper, bifurcation and stability of two kinds of constant stationary solutions for non-reversible amplitude equations on a bounded domain with Neumann boundary conditions are investigated by using the perturbation theory and weak nonlinear analysis. The asymptotic behaviors and local properties of two explicit steady state solutions, and pitch-fork bifurcations are also obtained if the bounded domain is regarded as a parameter. In addition, the stability of a new increasing or decaying local steady state solution with oscillations are analyzed.

MSC:

35B32 Bifurcations in context of PDEs
35B35 Stability in context of PDEs
35K51 Initial-boundary value problems for second-order parabolic systems
35K57 Reaction-diffusion equations
37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
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[1] I. S. Aranson and L. Kramer,The world of the complex Ginzburg-Landau equation, Rev. Mod. Phys, 2002, 74(1), 99-143. · Zbl 1205.35299
[2] D. J. Benny,Long waves in liquid film, J. Math. Phys, 1966, 45, 150-155. · Zbl 0148.23003
[3] D. Bl¨omker and W. W. Mohammed,Amplitude equations for SPDEs with cubic nonlinearities, Stochastics, Int. J. Probab. Stoch. Process, 2013, 85(2), 181- 215. · Zbl 1291.60127
[4] J. Burke, S. M. Houghton and E. Knobloch,Swift-Hohenberg equation with broken reflection symmetry, Phys. Rev. E, 2009, 80, 036202.
[5] P. Coullet and G. Iooss,Instabilities of one-dimensional cellular patterns, Phys. Rev. Lett, 1990, 64(8), 866-869. · Zbl 1050.82518
[6] S. M. Cox and P. C. Matthews,New instabilities of two-dimensional rotating convection and magnetoconvection, Physica D, 2001, 149(3), 210-229. · Zbl 0974.35091
[7] J. H. P. Dawes,Localized pattern formation with a large-scale mode: slanted snaking, SIAM J. Appl. Dyn. Syst., 2008, 7(1), 186-206. · Zbl 1159.76333
[8] A. Doelman, R. A. Gardner and T. J. Kaper,Large stable pulse solutions in reaction-diffusion equations, Indiana Univ. Math. J., 2001, 50(1), 421-425. · Zbl 0994.35058
[9] A. Doelman, R. A. Gardner and T. J. Kaper,A stability index analysis of 1-D patterns of the Gray-Scott model, Mem. Amer. Math. Soc., 2002, 155, 737. · Zbl 0994.35059
[10] A. Doelman, G. Hek and N. Valkhoff,Stabilization by slow diffusion in a real Ginzburg-Landau system, J. Nonlin. Sci., 2004, 14(3), 237-278. · Zbl 1136.37356
[11] S. Fauve,Pattern forming instabilities, Hydrodynamics and Nonlinear Instabilities, eds. Godr´eche, C. and Manneville, P. (Cambridge University Press), 1998, 387-491. · Zbl 0904.76026
[12] G. Fibich and D. Shpigelman,Positive and necklace solitary waves on bounded domains, Physica D, 2016, 315, 13=-32. · Zbl 1364.35333
[13] M. Ghergu,Steady-state solutions for Gierer-Meinhardt type systems with Dirichlet boundary condition, Transactions of the American Mathematical Society, 2009, 361(8), 3953-3976. · Zbl 1172.35018
[14] K. Klepel, K. Bl¨omker and W. W. Mohammed,Amplitude equation for the generalized Swift Hohenberg equation with noise, ZAMP-Zeitschrift fur angewandte Mathematik und Physik, 2014, 65, 1107-1126. · Zbl 1322.60117
[15] C. P. Li and G. Chen,Bifurcation from an equilibrium of the steady state Kuramoto-Sivashinsky equation in two spatial dimensions, Int. J. Bifurcation and Chaos, 2011, 12(12), 103-114. · Zbl 1042.35021
[16] Y. Li,Hopf bifurcation in general systems of Brusselator type, Nonlinear Anal. R. W. A., 2016, 28, 32-47. · Zbl 1329.35053
[17] P. C. Matthews and S. M. Cox,One dimensional pattern formation with Galilean invariance near a stationary bifurcation, Phys. Rev. E, 2002, 62, 1-4.
[18] P. C. Matthews and S. M. Cox,Pattern formation with a conservation law, Nonlinearity, 2000, 13(4), 1293-1320. · Zbl 0960.35007
[19] A. Mielke,The Ginzburg-Landau equation in its role as a modulation equation, Handbook of Dynamical Systems 2 (North-Holland, Amsterdam), 2002, pp. 759-834. · Zbl 1041.37037
[20] J. Norbury, M. Winter and J. Wei,Existence and stability of singular patterns in a Ginzburg-Landau equation coupled with a mean field, Nonlinearity, 2002, 15(6), 2077-2096. · Zbl 1021.35010
[21] L. A. Peletier and V. Rottschafer,Pattern selection of solutions of the SwiftHohenberg equation, Physica D, 2004, 194(1-2), 95-126. · Zbl 1052.35076
[22] L. A. Peletier and J. F. Williams,Some canonical bifurcations in the SwiftHohenberg equation, SIAM J. Appl. Dyn. Syst., 2007, 6(1), 208-235. · Zbl 1210.34028
[23] J. B. Swift and P. C. Hohenberg,Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 1977, 15(1), 319-328.
[24] L. Shi and H. J. Gao,Bifurcation analysis of an amplitude equation, Int. J. Bifurcation and Chaos., 2013, 23(5), 1350081. · Zbl 1270.34031
[25] B. Sandstede and Y. C. Xu,Snakes and isolas in non-reversible conservative systems, Dynamical Systems, 2012, 27(3), 317-329. · Zbl 1254.35121
[26] A.
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