Pulse-like Spatial Patterns Described by Higher-Order Model Equations

Journal of Differential Equations

Volumn 150, Issue 1, 1998, Pages 124-187

  Peletier, L A ^a   Rotariu Bruma, A I ^a   Troy, W C ^b  

^a LEIDEN UNIVERSITY   (Netherlands)

^b UNIVERSITY OF PITTSBURGH   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001378357     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.1998.3480     Document Type: Article

References (32)

1. Akhmediev N. N., Buryak A. V., Karlsson M. Radiationless optical solitons with oscillating tails. Opt. Comm. 110:1994;540-544.
2. Amick C. J., Toland J. F. Homoclinic orbits in the dynamic phase space analogy of an elastic strutt. European J. Appl. Math. 3:1992;97-114.
3. Aronson D. G., Weinberger H. Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30:1978;33-76.
4. G. J. B. van den Berg, M.S. Thesis, Leiden University, 1996.
5. van den Berg G. J. B. Uniqueness of solutions of the extended Fisher-Kolmogorov equation. C. R. Acad. Sci. Paris. 326:1998;447-452.
6. Buffoni B., Champneys A. R., Toland J. F. Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system. J. Dynamics and Differential Equations. 8:1996;221-279.
7. Bodenschatz E., Kaiser M., Kramer L., Pesch W., Weber A., Zimmerman W. Patterns and defects in liquid crystals. New Trends in Nonlinear Dynamics and Pattern Forming Phenomena: The Geometry of Nonequilibruim. NATO ASI Series, Plenum. 1990;. p. 111.
8. Buffoni B., Toland J. F. Global existence of homoclinic periodic orbits for a class of autonomous Hamiltonian systems. J. Differential Equations. 18:1995;104-120.
9. Champneys A. R., Groves M. D. A global investigation of solitary-wave solutions to a two-parameter model for water waves. J. Fluid Mech. 342:1997;199-229.
10. Coullet P., Elphick C., Repaux D. Nature of spatial chaos. Phys. Rev. Lett. 58:1987;431-434.
11. Cross M. C., Hohenberg P. C. Pattern formation outside of equilibrium. Rev. Mod. Phys. 65:1977;851-1112.
12. Devaney R. Blue sky catastrophies in reversible and Hamiltonian systems. Indiana Univ. Math. J. 26:1977;247-263.
13. Dee G. T., van Saarloos W. Bistable systems with propagating fronts leading to pattern formation. Phys. Rev. Lett. 60:1988;2641-2644.
14. Fife P. C., McLeod J. B. The approach of solutions of nonlinear diffusion equations to travelling wave solutions. Arch. Rational Mech. Anal. 65:1977;335-362.
15. Gardner R. A., Jones C. K. R. T. Traveling waves of a perturbed diffusion equation arising in a phase field model. Indiana Univ. Math. J. 38:1989;1197-1222.
16. Hofer H., Toland J. Homoclinic, heteroclinic, and periodic orbits for a class of indefinite Hamiltonian systems. Math. Ann. 268:1984;387-403.
17. Hornreich R. M., Luban M., Shtrikman S. Critical behaviour at the onset ofkλ Phys. Rev. Lett. 35:1975;1678.
18. Kalies W. D., Kwapisz J., van der Vorst R. C. A. M. Homotopy classes for stable connections between Hamiltonian saddle-focus equilibria. Comm. Math. Phys. 193:1998;337-371.
19. Kalies W. D., van der Vorst R. C. A. M. Countable families of homoclinic and heteroclinic connections for the Extended Fisher-Kolmogorov equation. J. Differential Equations. 131:1996;209-228.
20. Kichenessamy S. Existence of solitary waves for water-wave models. Nonlinearity. 10:1997;133-151.
21. Kolmogorov A., Petrovski I., Piscounov N. Etude de l'quation de la difussion avec croissance de la quantité matière et son application à un problème biologique. Bull. Univ. Moskou Ser. Internat. Sect. A. 1:1937;1-25.
22. Powell J. A., Newell A. C., Jones C. K. R. T. Competition between generic and non-generic fronts in envelope equations. Phys. Rev. A. 44:1991;3636-3652.
23. L. A. Peletier, A. I. Rotariu-Bruma.
24. Peletier L. A., Troy W. C. Spatial patterns described by the Extended Fisher-Kolmogorov (EFK) equation: Kinks. Differential and Integral Equations. 8:1995;1279-1304.
25. Peletier L. A., Troy W. C. Spatial patterns described by the Extended Fisher-Kolmogorov (EFK) equation: Periodic solutions. SIAM J. Math. Anal. 28:1997;1317-1353.
26. Peletier L. A., Troy W. C. Chaotic solutions of the Extended Fisher-Kolmogorov (EFK) equation. J. Differential Equations. 129:1996;458-508.
27. Peletier L. A., Troy W. C. A topological shooting method and the existence of kinks of the Extended Fisher-Kolmogorov equation. Topol. Methods Nonlinear Anal. 6:1996;331-355.
28. Peletier L. A., Troy W. C., van der Vorst R. C. A. M. Stationary solutions of a fourth order nonlinear diffusion equation. Differential Equations. 35:1995;301-314.
29. Protter M. H., Weinberger H. F. Maximum Principles in Differential Equations. 1984;Springer-Verlag, Berlin/New York.
30. van Saarloos W. Front propagation into unstable states: Marginal stability as a dynamical mechanism for velocity selection. Phys. Rev. A. 37:1988;211-229.
31. Vanderbauwhede A., Fiedler B. Homoclinic period blowup in reversible and conservative systems. Z. angew. Math. Phys. 43:1992;292-318.
32. Zimmerman W. Propagating fronts near a Lifschitz point. Phys. Rev. Lett. 66:1991;1546.