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 |
Keywords:
Galerkin method; numerical examples; quintic Swift-Hohenberg equation; localized patterns; snaky bifurcation structure; rigorous numerics; Conley index; dynamical systems; numerical verificationReferences:
| [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 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
