The Evans function for nonlocal equations

Indiana University Mathematics Journal

Volumn 53, Issue 4, 2004, Pages 1095-1126

  Kapitula, Todd ^a   Kutz, Nathan ^b   Sandstede, Björn ^c  

^a UNIVERSITY OF NEW MEXICO   (United States)

^b UNIVERSITY OF WASHINGTON   (United States)

^c OHIO STATE UNIVERSITY   (United States)

Author keywords

Evans function; Master modelocking equation; Nonlocal equations

Indexed keywords

EID: 8844238493     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.2004.53.2431     Document Type: Article

References (30)

1. J.C. ALEXANDER, ROBERT A. GARDNER, and C.K.R.T. JONES, A topological invariant arising in the stability analysis of travelling waves, J. Reine Angew. Math. 410 (1990), 167-212. MR 92d:58028
2. AMITABHA BOSE and GREGORY A. KRIEGSMANN, Stability of localized structures in non-local reaction-diffusion equations, Methods Appl. Anal. 5 (1998), 351-366. MR 2000b:35117
3. _, Large amplitude solutions of spatially non-homogeneous non-local reaction diffusion equations, Methods Appl. Anal. 7 (2000), 295-311. MR2002k:35164
4. ARJEN DOELMAN, ROBERT A. GARDNER, and TASSO J. KAPER, A stability index analysis of 1-D patterns of the Gray-Scott model, Memoirs Amer. Math. Soc., vol. 155, 2002. 1 878 337
5. ROBERT A. GARDNER and C. K. R. T. JONES, Traveling waves of a perturbed diffusion equation arising in a phase field model, Indiana Univ. Math. J. 39 (1990), 1197-1222. MR 92d:35039
6. ROBERT A. GARDNER and KEVIN ZUMBRUN, The gap lemma and geometric criteria for instability of viscous shock profiles, Comm. Pure Appl. Math. 51 (1998), 797-855, http://dx.doi.org/10.1002/(SICI)1097-0312(199807)51:7¡797::AID- CPA3¿3.0.CO;2-1. MR 99c:35152
7. MARTIN GOLUBITSKY and DAVID G. SCHAEFFER, Singularities and groups in bifurcation theory. Vol. I, Applied Mathematical Sciences, vol. 51, Springer-Verlag, New York, 1985, ISBN 0-387-90999-0. MR 86e:58014
8. TODD KAPITULA, The Evans function and generalized Melnikov integrals, SIAM J. Math. Anal. 30 (1999), 273-297 (electronic), http://dx.doi.org/10-1137/ S0036141097327963. MR 99m:35120
9. T. KAPITULA, J.N. KUTZ, and B. SANDSTEDE, Stability of pulses in the master mode-locking equation, J. Opt. Soc. Amer. B 19 (2002), 740-746.
10. TODD KAPITULA and JONATHAN RUBIN, Existence and stability of standing hole solutions to complex Ginzburg-Landau equations, Nonlinearity 13 (2000), 77-112, http://dx.doi.org/10.1088/0951-7715/13/1/305. MR 2000m:35166
11. TODD KAPITULA and BJÖRN SANDSTEDE, Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations, Phys. D 124 (1998), 58-103. MR 99h:35199
12. _, Instability mechanism for bright solitary-wave solutions to the cubic-quintic Ginzburg-Landau equation, J. Opt. Soc. Amer. B Opt. Phys. 15 (1998), 2757-2762. MR 2000c:78023
13. _, Edge bifurcations for near integrable systems via Evans function techniques, SIAM J. Math. Anal. 33 (2002), 1117-1143 (electronic), http://dx.doi.org/10.1137/S0036141000372301. MR 2003c:37109
14. T. KAPITULA and B. SANDSTEDE, Eigenvalues and resonances using the Evans function, Discrete Cont. Dynam. Syst. B 10 (2004), 857-869.
15. TOSIO KATO, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, vol. 132, Springer-Verlag New York, Inc., New York, 1966. MR 34 #3324
16. J.N. KUTZ, B.C. COLLINGS, B. BERGMAN, S. TSUDA, S. CUNDIFF, W. KNOX, P. HOLMES, and M. WEINSTEIN, Modelocking pulse dynamics in a fibre laser with a saturable Bragg reflector, J. Opt. Soc. Amer. B 14 (1997).
17. ALEXANDER MIELKE, PHILIP HOLMES, and J. NATHAN KUTZ, Global existence and uniqueness for an optical fibre laser model, Nonlinearity 11 (1998), 1489-1504, http://dx.doi.org/10.1088/0951-7715/11/6/004. MR 99f:35194
18. K.J. PALMER, Exponential dichotomies and transversal homoclinic points, J. Diff. Eqns. 55 (1984), 225-256.
19. KENNETH J. PALMER, Exponential dichotomies and Fredholm operators, Proc. Amer. Math. Soc. 104 (1988), 149-156. MR 89k:34052
20. DANIELA PETERHOF, BJÖRN SANDSTEDE, and ARND SCHEEL, Exponential dichotomies for solitary-wave solutions of semilinear elliptic equations on infinite cylinders, J. Differential Equations 140 (1997), 266-308, http://dx.doi.org/10.1006/jdeq.1997.3303. MR 99b:35057
21. BJÖRN SANDSTEDE, Stability of multiple-pulse solutions, Trans. Amer. Math. Soc. 350 (1998), 429-472. MR 98d:35100
22. _, Stability of travelling waves, Handbook of Dynamical Systems, Vol. 2, North-Holland, Amsterdam, 2002, pp. 983-1055. MR 2004e:37121
23. B. SANDSTEDE and A. SCHEEL, Essential instability of pulses and bifurcations to modulated travelling waves, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), 1263-1290. MR 2001k:37127
24. B. SANDSTEDE and A. SCHEEL, Spectral stability of modulated travelling waves bifurcating near essential instabilities, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), 419-448. MR 2001j:35164
25. BJÖRN SANDSTEDE and ARND SCHEEL, Absolute and convective instabilities of waves on unbounded and large bounded domains, Phys. D 145 (2000), 233-277, http://dx.doi.org/10.1016/S0167-2789(00)00114-7. MR 2002k:37157
26. B. SANDSTEDE and A. SCHEEL, Evans function and blow-up methods in critical eigenvalue problems, Discrete Cont. Dynam. Syst. B 10 (2004), 941-964.
27. _, Absolute instabilities of standing pulses (2003) (preprint).
28. Michael I. WEINSTEIN, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), 472-491. MR 861:35130
29. LINGHAI ZHANG, Evans functions and asymptotic stability of traveling wave solutions, Chinese Ann. Math. Ser. B 22 (2001), 343-360. MR 2002d:35109
30. _, On stability of traveling wave solutions in synaptically coupled neuronal networks, Differential Integral Equations 16 (2003), 513-536. MR 2004c:45016