Stability of the static spike autosolitons in the Gray-Scott model

SIAM Journal on Applied Mathematics

Volumn 62, Issue 5, 2002, Pages 1463-1487

  Muratov, C B ^a   Osipov, V V ^a  

^a NEW JERSEY INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Pattern formation; Reaction diffusion systems; Self organization; Singular perturbation theory

Indexed keywords

ASYMPTOTIC STABILITY; CATALYSIS; CHEMICAL REACTIONS; DIFFUSION; DYNAMICS; PERTURBATION TECHNIQUES;

STATIC SPIKE AUTOSOLITONS;

SOLITONS;

EID: 0036588860     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139901384285     Document Type: Article

References (41)

1. A. Bose and G.A. Kriegsmann, Stability of localized structures in non-local reaction-diffusion equations, Methods Appl. Anal., 5 (1998), pp. 351-366.
2. M.C. Cross and P.S. Hohenberg, Pattern formation outside of equilibrium, Rev. Modern Phys., 65 (1993), pp. 851-1112.
3. J.D. Dockery and J.P. Keener, Diffusive effects on dispersion in excitable media, SIAM J. Appl. Math., 49 (1989), pp. 539-566.
4. A. Doelman, R.A. Gardner, and T.J. Kaper, Stability analysis of singular patterns in the 1D Gray-Scott model: A matched asymptotics approach, Phys. D, 122 (1998), pp. 1-36.
5. A. Doelman, T.J. Kaper, and P. Zegeling, Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity, 10 (1997), pp. 523-563.
6. A.L. Dubitskii, B.S. Kerner, and V.V. Osipov, Two types of spike autosolitons, Soviet Phys. Dokl., 34 (1989), pp. 906-908.
7. R.J. Field and M. Burger, EDS., Oscillations and Traveling Waves in Chemical Systems, Wiley, New York, 1985.
8. P. Gray and S. Scott, Autocatalytic reactions in the isothermal continuous stirred tank reactor, Chem. Engrg. Sci., 38 (1983), pp. 29-43.
9. J.K. Hale, L.A. Peletier, and W.C. Troy, Exact homoclinic and heteroclinic solutions of the Gray-Scott model for autocatalysis, SIAM J. Appl. Math., 61 (2000), pp. 102-130.
10. R. Kapral and K. Showalter, Chemical Waves and Patterns, Kluwer, Dordrecht, 1995.
11. B.S. Kerner and V.V. Osipov, Nonlinear theory of stationary strata in dissipative systems, Sov. Phys. - JETP, 47 (1978), pp. 874-885.
12. B.S. Kerner and V.V. Osipov, Striations in a heated electron-hole plasma, Sov. Phys. Semicond., 13 (1979), pp. 424-431.
13. B.S. Kerner and V.V. Osipov, Stochastically inhomogeneous structures in nonequilibrium systems, Sov. Phys. - JETP, 52 (1980), pp. 1122-1132.
14. B.S. Kerner and V.V. Osipov, Stationary and traveling dissipative structures in active kinetic media, Mikroelektronika, 10 (1981), pp. 407-432 (in Russian).
15. B.S. Kerner and V.V. Osipov, Properties of stationary dissipative structures in mathematical models of morphogenesis, Biophysics (USSR), 27 (1982), pp. 138-143.
16. B.S. Kerner and V.V. Osipov, Pulsating heterophase regions in nonequilibrium systems, Sov. Phys. - JETP, 56 (1982), pp. 1275-1282.
17. B.S. Kerner and V.V. Osipov, Traveling heterophase regions in nonequilibrium systems, Mikroelektronika, 12 (1983), pp. 512-529 (in Russian).
18. B.S. Kerner and V.V. Osipov, Highly nonequilibrium localized states in systems slightly away from thermodynamic equilibrium, Sov. Phys. - JETP Lett., 41 (1985), pp. 473-475.
19. B.S. Kerner and V.V. Osipov, Autosolitons in a hot semiconductor plasma, Sov. Phys. JETP, 62 (1985), pp. 337-348.
20. B.S. Kerner and V.V. Osipov, Autosolitons in active systems with diffusion, in Nonlinear Irreversible Processes, W. Ebeling and H. Ulbricht, eds., Springer-Verlag, Berlin, 1986.
21. B.S. Kerner and V.V. Osipov, Autosolitons, Sov. Phys. - Usp., 32 (1989), pp. 101-138.
22. B.S. Kerner and V.V. Osipov, Self-organization in active distributed media, Sov. Phys. - Usp., 33 (1990), pp. 679-719.
23. B.S. Kerner and V.V. Osipov, Autosolitons: A New Approach to Problem of Self-Organization and Turbulence, Kluwer, Dordrecht, 1994.
24. L.D. Landau and E.M. Lifshitz, Course of Theoretical Physics, Vol. 3, Pergamon Press, Oxford, UK, 1965.
25. A.S. Mikhailov, Foundations of Synergetics, Springer-Verlag, Berlin, 1990.
26. C.B. Muratov, Self-replication and splitting of domain patterns in reaction-diffusion systems with the fast inhibitor, Phys. Rev. E, 54 (1996), pp. 3369-3376.
27. C.B. Muratov and V.V. Osipov, General theory of instabilities for patterns with sharp interfaces in reaction-diffusion systems, Phys. Rev. E, 53 (1996), pp. 3101-3116.
28. C.B. Muratov and V.V. Osipov, Scenarios of domain pattern formation in a reaction-diffusion system, Phys. Rev. E, 54 (1996), pp. 4860-4879.
29. C.B. Muratov and V.V. Osipov, Spike Autosolitons in the Gray-Scott Model, CAMS Rep. 9900-10, NJIT, Newark, NJ (available at LANL archive: part-sol/9804001).
30. C.B. Muratov and V.V. Osipov, Static spike autosolitons in the Gray-Scott model, J. Phys. A, 33 (2000), pp. 8893-8916.
31. C.B. Muratov and V.V. Osipov, Traveling spike autosolitons in the Gray-Scott model, Phys. D, 155 (2001), pp. 112-131.
32. C.B. Muratov and V.V. Osipov, Spike autosolitons and pattern formation scenarios in the two-dimensional Gray-Scott model, European Phys. J.B., (2001), pp. 213-221.
33. J.D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989.
34. F.J. Niedernostheide, ED., Nonlinear Dynamics and Pattern Formation in Semiconductors and Devices, Springer-Verlag, Berlin, 1994.
35. V.V. Osipov, Criteria of spontaneous interconversions of traveling and static arbitrary-dimensional dissipative structures, Phys. D, 93 (1996), pp. 143-156.
36. V.V. Osipov and A.V. Severtsev, Theory of self-replication and granulation of spike autosolitons, Phys. Lett. A, 222 (1996), pp. 400-404.
37. V.V. Osipov and A.V. Severtsev, Transverse instability of spike autosolitons, Phys. Lett. A, 227 (1997), pp. 61-66.
38. J.E. Pearson, Complex patterns in a simple system, Science, 261 (1993), pp. 189-192.
39. W.N. Reynolds, J.E. Pearson, and S. Ponce-Dawson, Dynamics of self-replicating patterns in reaction-diffusion systems, Phys. Rev. Lett., 72 (1994), pp. 2797-2800.
40. W.N. Reynolds, J.E. Pearson, and S. Ponce-Dawson, Self-replicating spots in reaction-diffusion systems, Phys. Rev. E, 56 (1997), pp. 185-198.
41. J.C. WEI, Pattern formation in two-dimensional Gray-Scott model: Existence of single-spot solutions and their stability, Phys. D, 148 (2001), pp. 20-48.