Nonlinear convective stability of travelling fronts near Turing and Hopf instabilities

Journal of Differential Equations

Volumn 246, Issue 11, 2009, Pages 4371-4390

  Beck, Margaret ^a   Ghazaryan, Anna ^b   Sandstede, Björn ^a  

^a UNIVERSITY OF SURREY   (United Kingdom)

^b University of North Carolina at Chapel Hill   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 64149094353     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2009.02.003     Document Type: Article

References (21)

1. Gallay T., Schneider G., and Uecker H. Stable transport of information near essentially unstable localized structures. Discrete Contin. Dyn. Syst. Ser. B 4 2 (2004) 349-390
2. A. Ghazaryan, Nonlinear stability of high Lewis number combustion fronts, Indiana Univ. Math. J. (2009), in press
3. A. Ghazaryan, Y. Latushkin, S. Schecter, A.J. de Souza, Stability of gasless combustion fronts in one-dimensional solids, preprint, 2008
4. Ghazaryan A., and Sandstede B. Nonlinear convective instability of Turing-unstable fronts near onset: A case study. SIAM J. Appl. Dyn. Syst. 6 2 (2007) 319-347
5. van Harten A. On the validity of the Ginzburg-Landau equation. J. Nonlinear Sci. 1 4 (1991) 397-422
6. Henry D. Geometric Theory of Semilinear Parabolic Equations (1981), Springer-Verlag, Berlin
7. Kay A.L., and Sherratt J.A. On the persistence of spatiotemporal oscillations generated by invasion. IMA J. Appl. Math. 63 2 (1999) 199-216
8. Kirrmann P., Schneider G., and Mielke A. The validity of modulation equations for extended systems with cubic nonlinearities. Proc. Roy. Soc. Edinburgh Sect. A 122 (1992) 85-91
9. Mielke A. Bounds for the solutions of the complex Ginzburg-Landau equation in terms of the dispersion parameters. Phys. D 117 1-4 (1998) 106-116
10. Mielke A. The Ginzburg-Landau equation in its role as a modulation equation. Handbook of Dynamical Systems II (2002), North-Holland, Amsterdam 759-834
11. Mielke A., and Schneider G. Attractors for modulation equations on unbounded domains-Existence and comparison. Nonlinearity 8 5 (1995) 743-768
12. Sandstede B. Stability of travelling waves. Handbook of Dynamical Systems II (2002), North-Holland, Amsterdam 983-1055
13. Sandstede B., and Scheel A. Absolute and convective instabilities of waves on unbounded and large bounded domains. Phys. D 145 3-4 (2000) 233-277
14. Sandstede B., and Scheel A. Essential instabilities of fronts: Bifurcation, and bifurcation failure. Dyn. Syst. 16 1 (2001) 1-28
15. Sandstede B., and Scheel A. Defects in oscillatory media: Toward a classification. SIAM J. Appl. Dyn. Syst. 3 (2004) 1-68
16. Schneider G. A new estimate for the Ginzburg-Landau approximation on the real axis. J. Nonlinear Sci. 4 1 (1994) 23-34
17. Schneider G. Error estimates for the Ginzburg-Landau approximation. Z. Angew. Math. Phys. 45 3 (1994) 433-457
18. Schneider G. Global existence via Ginzburg-Landau formalism and pseudo-orbits of Ginzburg-Landau approximations. Comm. Math. Phys. 164 1 (1994) 157-179
19. Schneider G. Hopf bifurcation in spatially extended reaction-diffusion systems. J. Nonlinear Sci. 8 1 (1998) 17-41
20. Sherratt J.A., Eagan B.T., and Lewis M.A. Oscillations and chaos behind predator-prey invasion: Mathematical artifact or ecological reality?. Philos. Trans. R. Soc. Lond. Ser. B 352 (1997) 21-38
21. Sherratt J.A. Invading wave fronts and their oscillatory wakes are linked by a modulated travelling phase resetting wave. Phys. D 117 1-4 (1998) 145-166