Global branches of multi-bump periodic solutions of the Swift-Hohenberg equation

Archive for Rational Mechanics and Analysis

Volumn 158, Issue 2, 2001, Pages 91-153

  Van Den Berg, G J B ^a   Peletier, L A ^a   Troy, W C ^b  

^a LEIDEN UNIVERSITY   (Netherlands)

^b UNIVERSITY OF PITTSBURGH   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035594306     PISSN: 00039527     EISSN: None     Source Type: Journal    
DOI: 10.1007/PL00004243     Document Type: Article

References (30)

1. G. J. B. VAN DEN BERG, MSc Thesis, Mathematical Institute, Leiden University, 1996.
2. G. J. B. VAN DEN BERG, Uniqueness of solutions for the extended Fisher-Kolmogorov equation, C. R. Acad. Sci. Paris 326, Série I (1998) 447-452.
3. G. J. B. VAN DEN BERG, The phase plane picture for a class of fourth-order conservative differential equations, J. Diff. Eqns. 161 (2000) 110-153.
4. B. BUFFONI, A. R. CHAMPNEYS & J. F. TOLAND, Bifurcation and coalescence of a plethora of homoclinic orbits for autonomous Hamiltonian systems, J. Dynamics and Differential Equations 8 (1996) 221-279.
5. M. C. CROSS & P. C. HOHENBERG, Pattern formation outside of equilibrium, Rev. Mod. Phys. 65 (1993) 851-1112.
6. P. COLLET & J. P. ECKMANN, Instabilities and fronts in extended systems, Princeton Series in Physics, Princeton University Press, 1990.
7. P. COULLET, C. ELPHICK & D. REPAUX, Nature of spatial chaos, Phys. Rev. Lett. 58 (1987) 431-434.
8. A. R. CHAMPNEYS & J. F. TOLAND, Bifurcation of a plethora of multi-modal homoclinic orbits for autonomous Hamiltonian systems, Nonlinearity 6 (1993) 665-722.
9. A. R. CHAMPNEYS & J. M. T. THOMPSON, A multiplicity of localised buckling modes for twisted rod equations, Proc. Royal Soc. of London A 452 (1996) 2467-2491.
10. B. D. COLEMAN, M. MARCUS & V. J. MIZEL, On the thermodynamics of periodic phases, Arch. Rational Mech. Anal. 117 (1992) 321-347.
11. E. J. DOEDEL et. al. , AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations, available by ftp from ftp.cs.concordia.ca
12. V. DANANIĆ & A. BJELIŠ, Domain patterns in incommensurate systems with the uniaxial real order parameter, Phys. Rev. E 50 (1994) 3900-3910.
13. G. T. DEE & W. VAN SAARLOOS, Bistable systems with propagating fronts leading to pattern formation, Phys. Rev. Lett. 60 (1988) 2641-2644.
14. G. B. ERMENTROUT, Phaseplane, the dynamical systems tool 3.0, Brooks/ cole Publ. co. Pacific Grove, CA, 1988.
15. J. GUGGENHEIMER & P. J. HOLMES, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Math. Sci., Springer Verlag, New York, 1983.
16. G. W. HUNT & A. BLACKMORE, Homoclinic and heteroclinic solutions of upheaval buckling, Phil. Trans. R. Soc. Lond. A. 355 (1997) 2185-2195.
17. W. D. KALIES, J. KWAPISZ & R. C. A. M. VAN DER VORST, Homotopy classes for stable connections between Hamiltonial saddle-focus equilibria, Commun. Math. Phys. 193 (1998) 337-371.
18. W. D. KALIES & R. C. A. M. VAN DER VORST, Multitransition homoclinic and heteroclinic solutions to the Extended Fisher-Kolmogorov equation, J.Diff. Eqns. 131 (1996) 209-228.
19. A. C. LAZER & P. J. MCKENNA, Large-amplitude oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev. 32 (1990) 537-578.
20. A. LEIZAROWITZ, & V. J. MIZEL, One dimensional infinite-horizon variational problems arising in continuum mechanics, Arch. Rational Mech. Anal. 106 (1989) 161-194.
21. V. J. MIZEL, L. A. PELETIER & W. C. TROY, Periodic phases in second order materials, Arch. Rational Mech. Anal. 145 (1998) 343-382.
22. L. A. PELETIER & W. C. TROY, Spatial patterns described by the Extended Fisher-Kolmogorov (EFK) equation: Kinks, Diff. Int. Eq. 8 (1995) 1279-1304.
23. L. A. PELETIER & W. C. TROY, Spatial patterns described by the Extended Fisher-Kolmogorov (EFK) equation: Periodic solutions, SIAM J. Math. Anal. 28 (1997) 1317-1353.
24. L. A. PELETIER & W. C. TROY, A topological shooting method and the existence of kinks of the Extended Fisher-Kolmogorov equation, Topol. Methods Nonlinear Analysis 6 (1996) 331-355.
25. L. A. PELETIER & W. C. TROY, Chaotic spatial patterns described by the Extended Fisher-Kolmogorov equation, J. Diff. Eqns. 126 (1996) 458-508.
26. L. A. PELETIER & W. C. TROY, Multi bump periodic travelling waves in suspension bridges, Proc. Roy. Soc. Edinb. 128A (1998) 631-659.
27. Y. POMEAU & P. MANNEVILLE, Wavelength selection in cellular flows, Phys. Letters 75A (1980) 296-298.
28. J. G. RAMSAY, A geologist's approach to rock deformation. In Inaugural Lectures -Imperial College of Science, Technology and Medicine, London, 1967, pp. 131-143.
29. J. B. SWIFT & P. C. HOHENBERG, Hydrodynamic Fluctuations at the convective instability, Phys. Rev. A 15 (1977) 319-328.
30. S. P. TIMOSHENKO & J. M. GERE, Theory of elastic stability, 2nd Edition, McGraw-Hill, New York, 1963.