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Rigorous numerics for localized patterns to the quintic Swift-Hohenberg equation. (English) Zbl 1067.65146
Summary: Localized patterns of the quintic Swift-Hohenberg equation are studied by bifurcation analysis and rigorous numerics. First of all, fundamental bifurcation structures around the trivial solution are investigated by a weak nonlinear analysis based on the center manifold theory. Then bifurcation structures with large amplitude are studied on Galerkin approximated dynamical systems, and a relationship between snaky branch structures of saddle-node bifurcations and localized patterns is discussed. Finally, a topological numerical verification technique proves the existence of several localized patterns as an original infinite-dimensional problem, which are beyond the local analysis.

MSC:
65P30 Numerical bifurcation problems
37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
37M20 Computational methods for bifurcation problems in dynamical systems
Software:
C-XSC; C-XSC 2.0
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[1] C. J. Budd, G.W. Hunt and R. Kuske, Asymptotics of cellular buckling close to the Maxwell load. Proc. R. Soc. Lond. A,457 (2001), 2935. · Zbl 1011.74023 · doi:10.1098/rspa.2001.0843
[2] C. Conley, Isolated Invariant Sets and the Morse Index. CBMS Lecture Notes38, A.M.S. Providence, R.I. 1978. · Zbl 0397.34056
[3] S. Day, Y. Hiraoka, K. Mischaikow and T. Ogawa, Rigorous numerics for global dynamics: a study of the Swift-Hohenberg equation. Submitted. · Zbl 1058.35050
[4] S. Day, O. Junge and K. Mischaikow, A rigorous numerical method for the global analysis of infinite dimensional discrete dynamical systems. In preparation. · Zbl 1059.37068
[5] L.Yu. Glebsky and L.M. Lerman, On small stationary localized solutions for the generalized 1-D Swift-Hohenberg equation. Chaos,5 (1995), 424. · Zbl 0952.37021 · doi:10.1063/1.166142
[6] Y. Hiraoka, T. Ogawa and K. Mischaikow, Conley index based numerical verification method for global bifurcations of the stationary solutions to the Swift-Hohenberg equation. Trans. Japan Soc. Indust. Appl. Math.,13, No. 2 (2003) 191.
[7] G. Iooss and M.C. Pérouéme, Perturbed homoclinic solutions in reversible 1:1 resonance vector fields. J. Diff. Eq.,102 (1993), 62. · Zbl 0792.34044 · doi:10.1006/jdeq.1993.1022
[8] H.B. Keller, Lectures on Numerical Methods in Bifurcation Problems. Springer-Verlag. Notes by A.K. Nandakumaran and Mythily Ramaswamy, Indian Institute of Science, Bangalore. 1987. · Zbl 0656.65063
[9] R. Klatte, U. Kulisch, A. Wiethoff, C. Lawo and M. Rauch, C-XSC: A C++ Class Library for Extended Scientific Computing. Springer-Verlag, 1993. · Zbl 0814.68035
[10] A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains–existence and comparison. Nonlinearity,8 (1995), 743. · Zbl 0833.35016 · doi:10.1088/0951-7715/8/5/006
[11] Y. Nishiura, Far-from-Equilibrium Dynamics. AMS Translations of Mathematical Monographs209, 2002. · Zbl 1013.37001
[12] S. Oishi and M. Rump, Fast verification of solutions of matrix equations. Numer. Math.,90 (2002), 755. · Zbl 0999.65015 · doi:10.1007/s002110100310
[13] S. Orszag, Numerical simulation of incompressible flows within simple boundaries. I. Galerkin (spectral) representations. Stud. Appl. Math.,50 (1971), 293. · Zbl 0237.76012
[14] H. Sakaguchi and H.R. Brand, Stable localized solutions of arbitrary length for the quintic Swift-Hohenberg equation. Physica D,97 (1996), 274. · doi:10.1016/0167-2789(96)00077-2
[15] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets. Trans. Amer. Math. Soc.,291 (1985), 1. · Zbl 0573.58020 · doi:10.1090/S0002-9947-1985-0797044-3
[16] A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions. Dynamics Reported125, Springer-Verlag, 1992. · Zbl 0751.58025
[17] M.K. Wadee, C.D. Coman and A.P. Bassom, Solitary wave interaction phenomena in a strut buckling model incorporating restabilisation. Physica D,163 (2002), 26. · Zbl 1082.74524 · doi:10.1016/S0167-2789(02)00350-0
[18] P.D. Woods and A.R. Champneys, Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian-Hopf bifurcation. Physica D,129 (1999), 147. · Zbl 0952.37009 · doi:10.1016/S0167-2789(98)00309-1
[19] P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation. Found. Comput. Math., 1 (2001), 255. · Zbl 0984.65101 · doi:10.1007/s002080010010
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