Alternate Evans functions and viscous shock waves

SIAM Journal on Mathematical Analysis

Volumn 32, Issue 5, 2001, Pages 929-962

  Benzoni Gavage, Sylvie ^a   Serre, Denis ^a   Zumbrun, Kevin ^b  

^a UNITÉ DE MATHÉMATIQUES PURES ET APPLIQUÉES   (France)

^b INDIANA UNIVERSITY   (United States)

Author keywords

Asymptotic stability; Traveling waves; Viscous conservation laws

Indexed keywords

EID: 0035538197     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141099361834     Document Type: Article

References (25)

1. J.C. ALEXANDER, R. GARDNER, AND C.K.R.T. JONES, A topological invariant arising in the stability analysis of travelling waves, J. Reine Angew. Math., 410 (1990), pp. 167-212.
2. J.C. ALEXANDER, M.G. GRILLAKIS, C.K.R.T. JONES, AND B. SANDSTEDE, Stability of pulses on optical fibers with phase-sensitive amplifiers, Z. Angew. Math. Phys., 48 (1997), pp. 175-192.
3. S. BENZONI-GAVAGE, On the stability of semidiscrete shock profiles by means of an Evans function in infinite dimension, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), pp. 377-382.
4. T.J. BRIDGES AND G. DERKS, Hodge duality and the Evans function, Phys. Lett. A, 251 (1999), pp. 363-372.
5. J.W. EVANS, Nerve axon equations. I. Linear approximations, Indiana Univ. Math. J., 21 (1971/1972), pp. 877-885.
6. J.W. EVANS, Nerve axon equations. II. Stability at rest, Indiana Univ. Math. J., 22 (1972/1973), pp. 75-90.
7. J.W. EVANS, Nerve axon equations. III. Stability of the nerve impulse, Indiana Univ. Math. J., 22 (1972/1973), pp. 577-593.
8. J.W. EVANS, Nerve axon equations. IV. The stable and the unstable impulse, Indiana Univ. Math. J., 24 (1974/1975), pp. 1169-1190.
9. H. FREISTÜHLER AND K. ZUMBRUN, Examples of Unstable Viscous Shock Waves, Technical report, Institut für Mathematik, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, 1998.
10. R.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), pp. 1197-1222.
11. R.A. GARDNER AND C.K.R.T. JONES, A stability index for steady state solutions of boundary value problems for parabolic systems, J. Differential Equations, 91 (1991), pp. 181-203.
12. R.A. GARDNER AND K. ZUMBRUN, The gap lemma and geometric criteria for instability of viscous shock profiles, Comm. Pure Appl. Math., 51 (1998), pp. 797-855.
13. D. HENRY, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981.
14. C.K.R.T. JONES, Stability of the travelling wave solution of the FitzHugh-Nagumo system, Trans. Amer. Math. Soc., 286 (1984), pp. 431-469.
15. C.K.R.T. JONES, R. GARDNER, AND T. KAPITULA, Stability of travelling waves for nonconvex scalar viscous conservation laws, Comm. Pure Appl. Math., 46 (1993), pp. 505-526.
16. T. KAPITULA, The Evans function and generalized Melnikov integrals, SIAM J. Math. Anal., 30 (1998), pp. 273-297.
17. T. KAPITULA AND B. SANDSTEDE, Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations, Phys. D, 124 (1998), pp. 58-103.
18. S. KAWASHIMA, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Ph.D. thesis, Kyoto University, Kyoto, 1983.
19. A. MAJDA AND R.L. PEGO, Stable viscosity matrices for systems of conservation laws, J. Differential Equations, 56 (1985), pp. 229-262.
20. R.L. PEGO AND M.I. WEINSTEIN, Eigenvalues, and instabilities of solitary waves, Philos. Trans. Roy. Soc. London Ser. A, 340 (1992), pp. 47-94.
21. D. SERRE, Systèmes de lois de conservation. I, in Hyperbolicité, entropies, ondes de choc, Diderot Editeur, Paris, 1996.
22. D. SERRE, La transition vers l'instabilité pour les ondes de choc multi-dimensionnelles, http:// www.umpa.ens-lyon.fr/ serre/PS/sw.ps (1999).
23. J. SWINTON, The stability of homoclinic pulses: A generalisation of Evans's method, Phys. Lett. A, 163 (1992), pp. 57-62.
24. K. ZUMBRUN AND P. HOWARD, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J., 47 (1998), pp. 741-871.
25. K. ZUMBRUN AND D. SERRE, Viscous and inviscid stability of multidimensional planar shock fronts, Indiana Univ. Math. J., 48 (1999), pp. 937-992.