Corner defects in almost planar interface propagation

Haragus, Mariana ; Scheel, Arnd

Annales de l'I.H.P. Analyse non linéaire, Tome 23 (2006), p. 283-329 / Harvested from Numdam

Détail

Publié le : 2006-01-01

MR 2217654 | Zbl 1098.35085 | 1 citation dans Numdam

DOI : https://doi.org/10.1016/j.anihpc.2005.03.003

Bibliographie

[1] Alexander J., Gardner R., Jones C.K.R.T., A topological invariant arising in the stability analysis of travelling waves, J. Reine Angew. Math. 410 (1990) 167-212. | MR 1068805 | Zbl 0705.35070

[2] Bardi M., Crandall M.G., Evans L.C., Soner H.M., Souganidis P.E., Viscosity Solutions and Applications, Lecture Notes in Math., vol. 1660, Springer-Verlag, Berlin, 1997. | MR 1462699

[3] Bonnet A., Hamel F., Existence of nonplanar solutions of a simple model of premixed Bunsen flames, SIAM J. Math. Anal. 31 (1999) 80-118. | MR 1742304 | Zbl 0942.35072

[4] Briggs R.J., Electron-Steam Interaction with Plasmas, MIT Press, Cambridge, 1964.

[5] Bykov V.V., The bifurcations of separatrix contours and chaos, Physica D 62 (1993) 290-299. | MR 1207427 | Zbl 0799.58054

[6] Caginalp G., Chen X., Convergence of the phase field model to its sharp interface limits, Eur. J. Appl. Math. 9 (1998) 417-445. | MR 1643668 | Zbl 0930.35024

[7] Cahn J.W., Mallet-Paret J., Van Vleck E.S., Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math. 59 (1999) 455-493. | MR 1654427 | Zbl 0917.34052

[8] Clarke S.R., Miller P.D., On the semi-classical limit for the focusing nonlinear Schrödinger equation: sensitivity to analytic properties of the initial data, Proc. Roy. Soc. London Ser. A (2002) 135-156. | MR 1879462 | Zbl 0997.35085

[9] Davies D.W., Blanchedeau P., Dulos E., De Kepper P., Dividing blobs, chemical flowers and patterned islands in a reaction-diffusion system, J. Chem. Phys. 102 (1998) 8236-8244.

[10] Doelman A., Sandstede B., Scheel A., Schneider G., Propagation of hexagonal patterns near onset, Eur. J. Appl. Math. 14 (2003) 85-110. | MR 1970238 | Zbl 1044.37049

[11] A. Doelman, B. Sandstede, A. Scheel, G. Schneider, The dynamics of modulated wave trains, Preprint, 2004.

[12] Engler H., Asymptotic stability of travelling wave solutions for perturbations with algebraic decay, J. Differential Equations 185 (2002) 348-369. | MR 1938123 | Zbl 1040.35094

[13] E. Eszter, Evans function analysis of the stability of periodic travelling wave solutions of the FitzHugh-Nagumo system, PhD Thesis, University of Massachusetts at Amherst, 1999.

[14] Fiedler B., Sandstede B., Scheel A., Wulff C., Bifurcation from relative equilibria of noncompact group actions: skew products, meanders, and drifts, Documenta Math. 1 (1996) 479-505. | MR 1425301 | Zbl 0870.58014

[15] Fife P.C., Dynamics of internal layers and diffusive interfaces, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 53, SIAM, Philadelphia, PA, 1988. | MR 981594 | Zbl 0684.35001

[16] Gardner R.A., Zumbrun K., The gap lemma and geometric criteria for instability of viscous shock profiles, Comm. Pure Appl. Math. 51 (1998) 797-855. | MR 1617251 | Zbl 0933.35136

[17] Golovin A.A., Nepomnyashchy A.A., Matkowsky B.J., Traveling and spiral waves for sequential flames with translation symmetry: coupled CGL-Burgers equations, Physica D 160 (2001) 1-28. | MR 1872551 | Zbl 0982.80007

[18] Hale J.K., Peletier L.A., Troy W.C., Stability and instability in the Gray-Scott model: the case of equal diffusivities, Appl. Math. Lett. 12 (1999) 59-65. | MR 1750599 | Zbl 0936.92034

[19] Hale J.K., Peletier L.A., Troy W.C., Exact homoclinic and heteroclinic solutions of the Gray-Scott model for autocatalysis, SIAM J. Appl. Math. 61 (2000) 102-130. | MR 1776389 | Zbl 0965.34037

[20] Hamel F., Monneau R., Solutions of semilinear elliptic equations in ${R}^{N}$ with conical-shaped level sets, Comm. Partial Differential Equations 25 (2000) 769-819. | MR 1759793 | Zbl 0952.35041

[21] Hamel F., Monneau R., Roquejoffre J.-M., Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Sci. École Norm. Sup. (4) 37 (2004) 469-506. | Numdam | MR 2060484 | Zbl 1085.35075

[22] Hamik C.T., Steinbock O., Shock structures and bunching fronts in excitable reaction-diffusion systems, Phys. Rev. E 65 (2002) 046224.

[23] Haragus M., Kirchgässner K., Breaking the dimension of a steady wave: some examples, in: Nonlinear Dynamics and Pattern Formation in the Natural Environment, Pitman Res. Notes Math. Ser., vol. 335, Longman, Harlow, 1995, pp. 119-129. | MR 1381910 | Zbl 0837.35131

[24] Haragus M., Kirchgässner K., Breaking the dimension of solitary waves, in: Progress in Partial Differential Equations: The Metz Surveys, 4, Pitman Res. Notes Math. Ser., vol. 345, Longman, Harlow, 1996, pp. 216-228. | MR 1394716 | Zbl 0856.35110

[25] Hartmann N., Kevrekidis Y., Imbihl R., Pattern formation in restricted geometries: the NO + CO reaction on Pt(100), J. Chem. Phys. 112 (2000) 6795-6803.

[26] Hastings S., On the existence of homoclinic and periodic orbits for the FitzHugh-Nagumo equations, Quart. J. Math. Oxford Ser. 27 (1976) 123-134. | MR 393759 | Zbl 0322.92008

[27] Henry D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 804, Springer-Verlag, New York, 1981. | MR 610244 | Zbl 0456.35001

[28] Hoff D., Zumbrun K., Asymptotic behavior of multidimensional scalar viscous shock fronts, Indiana Univ. Math. J. 49 (2000) 427-474. | MR 1793680 | Zbl 0967.76049

[29] Iibun T., Sakamoto K., Internal layers intersecting the boundary of domain in the Allen-Cahn equation, Japan J. Indust. Appl. Math. 18 (2001) 697-738. | MR 1862434 | Zbl 0987.35078

[30] Imbihl R., Engel H., Eiswirth M., Dynamics of patterns of chemical reactions on surfaces, in: Evolution of Spontaneous Structures in Dissipative Continuous Systems, Lecture Notes in Phys., vol. 55, Springer, Berlin, 1998, pp. 384-410. | Zbl 0933.76098

[31] Iooss G., Adelmeyer M., Topics in Bifurcation Theory and Applications, Adv. Ser. Nonlinear Dynam., vol. 3, World Scientific, River Edge, NJ, 1998. | MR 1695170 | Zbl 0968.34027

[32] Jones C.K.R.T., Stability of the travelling wave solution of the FitzHugh-Nagumo system, Trans. Amer. Math. Soc. 286 (1984) 431-469. | MR 760971 | Zbl 0567.35044

[33] Jones C.K.R.T., Gardner R., Kapitula T., Stability of travelling waves for nonconvex scalar viscous conservation laws, Comm. Pure Appl. Math. 46 (1993) 505-526. | MR 1211740 | Zbl 0791.35078

[34] Kærn M., Menzinger M., Pulsating wave propagation in reactive flows: flow-distributed oscillations, Phys. Rev. E 61 (2000) 3334-3338.

[35] Kapitula T., On the stability of travelling waves in weighted ${L}^{\infty }$ spaces, J. Differential Equations 112 (1994) 179-215. | MR 1287557 | Zbl 0803.35067

[36] Kapitula T., Multidimensional stability of planar travelling waves, Trans. Amer. Math. Soc. 349 (1997) 257-269. | MR 1360225 | Zbl 0866.35021

[37] Kapitula T., Sandstede B., Stability of bright solitary wave solutions to perturbed nonlinear Schrödinger equations, Physica D 124 (1998) 58-103. | MR 1662530 | Zbl 0935.35150

[38] Kent P., Elgin J., Travelling-waves of the Kuramoto-Sivashinsky equation: period-multiplying bifurcations, Nonlinearity 5 (1992) 899-919. | MR 1174223 | Zbl 0771.35006

[39] Kirchgässner K., Wave-solutions of reversible systems and applications, J. Differential Equations 45 (1982) 113-127. | MR 662490 | Zbl 0507.35033

[40] Lega J., Moloney J.V., Newell A.C., Swift-Hohenberg equation for lasers, Phys. Rev. Lett. 73 (1994) 2978-2981.

[41] Liu T.-P., Métivier G., Smoller J., Temple B., Yong W.-A., Zumbrun K., Advances in the Theory of Shock Waves, Progr. Nonlinear Differential Equations Appl., vol. 47, Birkhäuser, Boston, MA, 2001. | MR 1842773

[42] Lombardi E., Oscillatory Integrals and Phenomena beyond all Algebraic Orders. With Applications to Homoclinic Orbits in Reversible Systems, Lecture Notes in Math., vol. 1741, Springer-Verlag, Berlin, 2000. | MR 1770093 | Zbl 0959.34002

[43] Markus L., Quadratic differential equations and non-associative algebras, in: Contributions to the Theory of Nonlinear Oscillations, vol. V, Princeton University Press, Princeton, NJ, 1960, pp. 185-213. | MR 132743 | Zbl 0119.29803

[44] Matkowsky B.J., Olagunju D.O., Pulsations in a burner-stabilized premixed plane flame, SIAM J. Appl. Math. 40 (1981) 551-562. | MR 614749 | Zbl 0484.76094

[45] Mielke A., Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci. 10 (1988) 51-66. | MR 929221 | Zbl 0647.35034

[46] Nishiura Y., Ueyama D., Spatio-temporal chaos for the Gray-Scott model, Physica D 150 (2001) 137-162. | Zbl 0981.35022

[47] Palmer K.J., Exponential dichotomies and Fredholm operators, Proc. Amer. Math. Soc. 104 (1988) 149-156. | MR 958058 | Zbl 0675.34006

[48] Pismen L.M., Nepomnyashchy A.A., Propagation of the hexagonal pattern, Europhys. Lett. 27 (1994) 433-436.

[49] Sakamoto K., Invariant manifolds in singular perturbation problems for ordinary differential equations, Proc. Roy. Soc. Edinburgh Sect. A 116 (1990) 45-78. | MR 1076353 | Zbl 0719.34086

[50] Sandstede B., Scheel A., Absolute and convective instabilities of waves on unbounded and large bounded domains, Physica D 145 (2000) 233-277. | MR 1782392 | Zbl 0963.34072

[51] Sandstede B., Scheel A., On the structure of spectra of modulated travelling waves, Math. Nachr. 232 (2001) 39-93. | MR 1871473 | Zbl 0994.35025

[52] Sandstede B., Scheel A., Essential instabilities of fronts: bifurcation and bifurcation failure, Dynamical Systems 16 (2001) 1-28. | MR 1835905 | Zbl 1055.37069

[53] Sandstede B., Scheel A., On the stability of periodic travelling waves with large spatial period, J. Differential Equations 172 (2001) 134-188. | MR 1824088 | Zbl 0994.34035

[54] Sandstede B., Scheel A., Defects in oscillatory media: toward a classification, SIAM J. Appl. Dyn. Syst. 3 (2004) 1-68. | MR 2067899 | Zbl 1059.37062

[55] Sandstede B., Scheel A., Wulff C., Dynamics of spiral waves on unbounded domains using center-manifold reduction, J. Differential Equations 141 (1997) 122-149. | MR 1485945 | Zbl 0888.35053

[56] Sattinger D.H., On the stability of waves of nonlinear parabolic systems, Adv. in Math. 22 (1976) 312-355. | MR 435602 | Zbl 0344.35051

[57] Scheel A., Radially symmetric patterns of reaction-diffusion systems, Mem. Amer. Math. Soc. 165 (2003). | MR 1997690 | Zbl 1036.35092

[58] Schöll E., Nonlinear Spatio-Temporal Dynamics and Chaos in Semiconductors, Cambridge Nonlinear Science Series, vol. 10, Cambridge University Press, 2001.

[59] Sivashinsky G.I., Nonlinear analysis of hydrodynamic instability in laminar flames I. Derivation of basic equations, Acta Astronautica 4 (1977) 1177-1206. | MR 502829 | Zbl 0427.76047

[60] Tsujikawa T., Nagai T., Mimura M., Kobayashi R., Ikeda H., Stability properties of traveling pulse solutions of the higher-dimensional FitzHugh-Nagumo equations, Japan J. Appl. Math. 6 (1989) 341-366. | MR 1019681 | Zbl 0702.35017

[61] Tyson J.J., Keener J.P., Singular perturbation theory of traveling waves in excitable media (a review), Physica D 32 (1988) 327-361. | MR 980194 | Zbl 0656.76018

[62] Van Saarloos W., Front propagation into unstable states. II. Linear versus nonlinear marginal stability and rate of convergence, Phys. Rev. A 39 (1989) 6367-6390. | MR 1003567

[63] Van Saarloos W., Front propagation into unstable states, Phys. Rep. 386 (2003) 29-222. | Zbl 1042.74029

[64] Volpert A.I., Volpert V.A., Volpert V.A., Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monographs, vol. 140, American Mathematical Society, Providence, RI, 1994. | MR 1297766 | Zbl 1001.35060

[65] Williams F.A., Combustion Theory, Benjamin Cummings, Menlo Park, 1985.

[66] Winfree A.T., The Geometry of Biological Time, Interdiscip. Appl. Math., vol. 12, Springer-Verlag, New York, 2001. | MR 1833606 | Zbl 1014.92001

[67] Yanagida E., Stability of fast travelling pulse solutions of the FitzHugh-Nagumo equations, J. Math. Biology 22 (1985) 81-104. | MR 802737 | Zbl 0566.92009