Rigorous numerics for localized patterns to the quintic swift-hohenberg equation

Japan Journal of Industrial and Applied Mathematics

Volumn 22, Issue 1, 2005, Pages 57-75

  Hiraoka, Yasuaki ^a   Ogawa, Toshiyuki ^a  

^a OSAKA UNIVERSITY   (Japan)

Author keywords

Conley index; Localized patterns; Quintic Swift Hohenberg equation; Rigorous numerics; Snaky bifurcation structure

Indexed keywords

EID: 15944428231     PISSN: 09167005     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF03167476     Document Type: Article

References (19)

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.
2. C. Conley, Isolated Invariant Sets and the Morse Index. CBMS Lecture Notes 38, A.M.S. Providence, R.I. 1978.
3. S. Day, Y. Hiraoka, K. Mischaikow and T. Ogawa, Rigorous numerics for global dynamics: a study of the Swift-Hohenberg equation. Submitted.
4. S. Day, O. Junge and K. Mischaikow, A rigorous numerical method for the global analysis of infinite dimensional discrete dynamical systems. In preparation.
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.
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.
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.
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.
10. A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains - existence and comparison. Nonlinearity, 8 (1995), 743.
11. Y. Nishiura, Far-from-Equilibrium Dynamics. AMS Translations of Mathematical Monographs 209, 2002.
12. S. Oishi and M. Rump, Fast verification of solutions of matrix equations. Numer. Math., 90 (2002), 755.
13. S. Orszag, Numerical simulation of incompressible flows within simple boundaries. I. Galerkin (spectral) representations. Stud. Appl. Math., 50 (1971), 293.
14. H. Sakaguchi and H.R. Brand, Stable localized solutions of arbitrary length for the quintic Swift-Hohenberg equation. Physica D, 97 (1996), 274.
15. D. Salamon, Connected simple systems and the Conley index of isolated invariant sets. Trans. Amer. Math. Soc., 291 (1985), 1.
16. A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions. Dynamics Reported 125, Springer-Verlag, 1992.
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.
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.
19. P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation. Found. Comput. Math., 1 (2001), 255.