Slowly modulated two-pulse solutions in the Gray-Scott model I: Asymptotic construction and stability

SIAM Journal on Applied Mathematics

Volumn 61, Issue 3, 2000, Pages 1080-1102

  Doelman, Arjen ^a   Eckhaus, Wiktor ^b   Kaper, Tasso J ^c  

^a UNIVERSITY OF AMSTERDAM   (Netherlands)

^b UTRECHT UNIVERSITY   (Netherlands)

^c Boston University   (United States)

Author keywords

Modulated traveling waves; Reaction diffusion equations; Self replicating patterns; Singular perturbation theory

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; GEOMETRY; MATHEMATICAL MODELS; PERTURBATION TECHNIQUES; PROBLEM SOLVING; VELOCITY;

GRAY-SCOTT MODEL; NONLOCAL EIGENVALUE PROBLEM; TWO-PULSE SOLUTIONS;

ASYMPTOTIC STABILITY;

EID: 0034912344     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139999354923     Document Type: Article

References (22)

1. A. DOELMAN, W. ECKHAUS, AND T.J. KAPER, Slowly modulated two-pulse solutions in the Gray-Scott model II: Geometric theory, bifurcations, and splitting dynamics, SIAM J. Appl. Math., to appear.
2. A. DOELMAN, T.J. KAPER, AND P. ZEGELING, Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity, 10 (1997), pp. 523-563.
3. A. DOELMAN, R.A. GARDNER, AND T.J. KAPER, Stability analysis of singular patterns in the 1-D Gray-Scott model: A matched asymptotics approach, Phys. D, 122 (1998), pp. 1-36.
4. A. DOELMAN, R.A. GARDNER, AND T.J. KAPER, A stability index analysis of 1-D patterns of the Gray-Scott model, Mem. Amer. Math. Soc., to appear.
5. A. DOELMAN, R.A. GARDNER, AND T.J. KAPER, Large stable pulse solutions in reaction-diffusion equations, Indiana Univ. Math. J., 49 (2000).
6. W. ECKHAUS, Asymptotic Analysis of Singular Perturbations, North-Holland, Amsterdam, 1979.
7. W. ECKHAUS AND R. KUSKE, Pattern formation in systems with slowly varying geometry, SIAM J. Appl. Math., 57 (1997), pp. 112-152.
8. S. EI, Y. NISHIURA, AND B. SANDSTEDE, Pulse-interaction approach to self-replicating dynamics in reaction-diffusion systems, in preparation.
9. C. ELPHICK, A. HAGBERG, AND E. MERON, Dynamical front transitions and spiral vortex nucleation, Phys. Rev. E, 51 (1995), pp. 3052-3058.
10. P. GRAY AND S.K. SCOTT, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system A + 2B → 3B, B → C, Chem. Engrg. Sci., 39 (1984), pp. 1087-1097.
11. 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.
12. J.K. HALE, L.A. PELETIER, AND W.C. TROY, Stability and instability in the Gray-Scott model: The case of equal diffusivities, Appl. Math. Lett., 12 (1999), pp. 59-65.
13. K.J. LEE AND H.L. SWINNEY, Lamellar structures and self-replicating spots in a reaction-diffusion system, Phys. Rev. E, 51 (1995), pp. 1899-1915.
14. K.-J. LIN, W.D. MCCORMICK, J.E. PEARSON, AND H.L. SWINNEY, Experimental observation of self-replicating spots in a reaction-diffusion system, Nature, 369 (1994), pp. 215-218.
15. D.S. MORGAN, A. DOELMAN, AND T.J. KAPER, Stationary periodic orbits in the 1-D Gray-Scott model, Meth. Appl. Anal., 7 (2000), pp. 105-150.
16. C. MURATOV AND V. OSIPOV, Spike Autosolitons in the Gray-Scott Model, preprint, 1998.
17. Y. NISHIURA AND D. UEYAMA, A skeleton structure for self-replication dynamics, Phys. D, 130 (1999), pp. 73-104.
18. J.E. PEARSON, Complex patterns in a simple system, Science, 261 (1993), pp. 189-192.
19. V. PETROV, S.K. SCOTT, AND K. SHOWALTER, Excitability, wave reflection, and wave splitting in a cubic autocatalysis reaction-diffusion system, Philos. Trans. Roy. Soc. London, Ser. A, 347 (1994), pp. 631-642.
20. 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.
21. W.N. REYNOLDS, S. PONCE-DAWSON, AND J.E. PEARSON, Self-replicating spots in reaction-diffusion systems, Phys. Rev. E, 56 (1997), pp. 185-198.
22. A. VAN HARTEN, Nonlinear singular perturbation problems: Proofs of correctness of a formal approximation based on a contraction principle in a Banach space, J. Math. Anal. Appl., 65 (1978), pp. 126-168.