{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:31:03Z","timestamp":1787232663359,"version":"build-2736575974"},"reference-count":27,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>The three-component reaction-diffusion system introduced in [C. P. Schenk et al., Phys. Rev. Lett., 78 (1997), pp. 3781\u20133784] has become a paradigm model in pattern formation. It exhibits a rich variety of dynamics of fronts, pulses, and spots. The front and pulse interactions range in type from weak, in which the localized structures interact only through their exponentially small tails, to strong interactions, in which they annihilate or collide and in which all components are far from equilibrium in the domains between the localized structures. Intermediate to these two extremes sits the semistrong interaction regime, in which the activator component of the front is near equilibrium in the intervals between adjacent fronts but both inhibitor components are far from equilibrium there, and hence their concentration profiles drive the front evolution. In this paper, we focus on dynamically evolving N-front solutions in the semistrong regime. The primary result is use of a renormalization group method to rigorously derive the system of N coupled ODEs that governs the positions of the fronts. The operators associated with the linearization about the N-front solutions have N small eigenvalues, and the N-front solutions may be decomposed into a component in the space spanned by the associated eigenfunctions and a component projected onto the complement of this space. This decomposition is carried out iteratively at a sequence of times. The former projections yield the ODEs for the front positions, while the latter projections are associated with remainders that we show stay small in a suitable norm during each iteration of the renormalization group method. Our results also help extend the application of the renormalization group method from the weak interaction regime for which it was initially developed to the semistrong interaction regime. The second set of results that we present is a detailed analysis of this system of ODEs, providing a classification of the possible front interactions in the cases of $N=1,2,3,4$, as well as how front solutions interact with the stationary pulse solutions studied earlier in [A. Doelman, P. van Heijster, and T. J. Kaper, J. Dynam. Differential Equations, 21 (2009), pp. 73\u2013115; P. van Heijster, A. Doelman, and T. J. Kaper, Phys. D, 237 (2008), pp. 3335\u20133368]. Moreover, we present some results on the general case of N-front interactions.<\/jats:p>","DOI":"10.1137\/080744785","type":"journal-article","created":{"date-parts":[[2010,4,12]],"date-time":"2010-04-12T09:40:02Z","timestamp":1271065202000},"page":"292-332","source":"Crossref","is-referenced-by-count":44,"title":["Front Interactions in a Three-Component System"],"prefix":"10.1137","volume":"9","author":[{"given":"P.","family":"van Heijster","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"A.","family":"Doelman","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"T. J.","family":"Kaper","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"K.","family":"Promislow","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,4,9]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/s00222-008-0141-y"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(01)00360-8"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"W. A. Coppel,\n                      Dichotomies in Stability Theory\n                      , Lecture Notes in Math. 629, Springer-Verlag, Berlin, New York, 1978.","DOI":"10.1007\/BFb0067780"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/S1111111102405719"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/050646883"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1007\/s10884-008-9125-2"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1023\/A:1012980128575"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(02)00379-2"},{"key":"R9","doi-asserted-by":"crossref","first-page":"617","DOI":"10.3934\/dcds.2009.23.617","volume":"23","author":"Guha M.","year":"2009","journal-title":"Discrete Contin. Dynam. Syst.","ISSN":"https:\/\/id.crossref.org\/issn\/1078-0947","issn-type":"print"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"S. V. Gurevich, Sh. Amiranashvili, and H.G. Purwins,\n                      Breathing dissipative solitons in three-component reaction-diffusion system\n                      , Phys. Rev. E, 74 (2006), paper 066201.","DOI":"10.1103\/PhysRevE.74.066201"},{"key":"R11","unstructured":"L. H\u00f6rmander,\n                      The Analysis of Linear Partial Differential Operators.\n                      I.\n                      Distribution Theory and Fourier Analysis\n                      , Grundlehren Math. Wiss. 256, Springer-Verlag, Berlin, 1983."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/S0036139901393676"},{"key":"R13","unstructured":"S. Lefschetz,\n                      Differential Equations: Geometric Theory\n                      , Dover Publications, New York, 1977."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2008.06.015"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.14492\/hokmj\/1285766659"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"Y. Nishiura, T. Teramoto, and K.I. Ueda,\n                      Scattering and separators in dissipative systems\n                      , Phys. Rev. E, 67 (2003), paper 056210.","DOI":"10.1103\/PhysRevE.67.056210"},{"key":"R17","doi-asserted-by":"crossref","unstructured":"Y. Nishiura, T. Teramoto, X. Yuan, and K.I. Ueda,\n                      Dynamics of traveling pulses in heterogeneous media\n                      , Chaos, 17 (2007), paper 037104.","DOI":"10.1063\/1.2778553"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.57.6432"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/S0036141000377547"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"B. Sandstede,\n                      Stability of travelling waves\n                      , in Handbook of Dynamical Systems 2, North\u2013Holland, Amsterdam, 2002, pp. 983\u20131055.","DOI":"10.1016\/S1874-575X(02)80039-X"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.78.3781"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/040620990"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1137\/0514066"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2008.07.014"},{"key":"R25","doi-asserted-by":"crossref","unstructured":"V. K. Vanag and I. R. Epstein,\n                      Localized patterns in reaction-diffusion systems\n                      , Chaos, 17 (2007), paper 037110.","DOI":"10.1063\/1.2752494"},{"key":"R26","doi-asserted-by":"crossref","unstructured":"X. Yuan, T. Teramoto, and Y. Nishiura,\n                      Heterogeneity-induced defect bifurcation and pulse dynamics for a three-component reaction-diffusion system\n                      , Phys. Rev. E, 75 (2007), paper 036220.","DOI":"10.1103\/PhysRevE.75.036220"},{"key":"R27","doi-asserted-by":"crossref","unstructured":"S. Zelik and A. Mielke,\n                      Multi-pulse evolution and space-time chaos in dissipative systems\n                      , Mem. Amer. Math. 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