[Approximation dʼondes progressives dans un référentiel non-local en mouvement]
Dans cette Note nous considérons le développement de méthodes permettant de préciser aussi bien les profils que la vitesse des ondes progressives pour des équations de réaction–diffusion, modélisées par des équations paraboliques semi-linéaires à une dimension dʼespace. Moyennant un changement de variable non-local, les profils deviennent des solutions stationnaires dʼun problème dʼévolution non-local. Nous démontrons que, dans cette nouvelle formulation, aussi bien les profils que les vitesses de propagation des ondes progressives deviennent des états stationnaires asymptotiques stables lorsque le temps tend vers lʼinfini. On analyse aussi la restriction de ce nouveau problème non-local de Cauchy en espace, à un intervalle dʼespace fini. Lorsque lʼintervalle dʼespace tronqué est assez grand on montre quʼil existe un état stationnaire unique et que si lʼintervalle tend vers la droite réelle toute entière, lʼétat stationnaire converge vers le profil de lʼonde progressive. Ceci permet de développer des méthodes numériques efficaces pour le calcul des profils et vitesses de ces ondes progressives nonlinéaires.
The profiles of traveling wave solutions of a 1-d reaction–diffusion parabolic equation are transformed into equilibria of a nonlocal equation, by means of an appropriate nonlocal change of variables. In this new formulation both the profile and the propagation speed of the traveling waves emerge as asymptotic limits of solutions of a nonlocal reaction–diffusion problem when time goes to infinity. In this Note we make these results rigorous analyzing the well-posedness and the stability properties of the corresponding nonlocal Cauchy problem. We also analyze its restriction to a finite interval with consistent boundary conditions. For large enough intervals we show that there is an asymptotically stable equilibrium which approximates the profile of the traveling wave in . This leads to efficient numerical algorithms for computing the traveling wave profile and speed of propagation.
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@article{CRMATH_2011__349_13-14_753_0,
author = {Arrieta, Jose M. and L\'opez-Fern\'andez, Maria and Zuazua, Enrique},
title = {On a nonlocal moving frame approximation of traveling waves},
journal = {Comptes Rendus. Math\'ematique},
pages = {753--758},
publisher = {Elsevier},
volume = {349},
number = {13-14},
year = {2011},
doi = {10.1016/j.crma.2011.07.001},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2011.07.001/}
}
TY - JOUR AU - Arrieta, Jose M. AU - López-Fernández, Maria AU - Zuazua, Enrique TI - On a nonlocal moving frame approximation of traveling waves JO - Comptes Rendus. Mathématique PY - 2011 SP - 753 EP - 758 VL - 349 IS - 13-14 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2011.07.001/ DO - 10.1016/j.crma.2011.07.001 LA - en ID - CRMATH_2011__349_13-14_753_0 ER -
%0 Journal Article %A Arrieta, Jose M. %A López-Fernández, Maria %A Zuazua, Enrique %T On a nonlocal moving frame approximation of traveling waves %J Comptes Rendus. Mathématique %D 2011 %P 753-758 %V 349 %N 13-14 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2011.07.001/ %R 10.1016/j.crma.2011.07.001 %G en %F CRMATH_2011__349_13-14_753_0
Arrieta, Jose M.; López-Fernández, Maria; Zuazua, Enrique. On a nonlocal moving frame approximation of traveling waves. Comptes Rendus. Mathématique, Tome 349 (2011) no. 13-14, pp. 753-758. doi : 10.1016/j.crma.2011.07.001. http://www.numdam.org/articles/10.1016/j.crma.2011.07.001/
[1] J.M. Arrieta, M. López-Fernández, E. Zuazua, Approximating traveling waves by equilibria of nonlocal equations, Preprint, 2011.
[2] Resolvent estimates for boundary value problems on large intervals via the theory of discrete approximations, Numer. Funct. Anal. Optim., Volume 28 (2007), pp. 603-629
[3] Freezing solutions of equivariant evolution equations, SIAM J. Appl. Dyn. Syst., Volume 3 (2004), pp. 85-116
[4] Phase conditions, symmetries, and PDE continuation, Numerical continuation methods for dynamical systems, Underst. Complex Syst., Springer, Dordrecht, 2007, pp. 301-330
[5] Spectral properties of non-local uniformly-elliptic operators, Electron. J. Differential Equations, Volume 126 (2006), p. 15
[6] Further spectral properties of uniformly elliptic operators that include a non-local term, Appl. Math. Comput., Volume 197 (2008), pp. 317-327
[7] The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rational Mech. Anal., Volume 65 (1977), pp. 335-361
[8] Non-local reaction diffusion equations, Fields Institute Communications, Volume 21 (1999), pp. 187-204
[9] Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer, Berlin, 1981
[10] The Evans function for nonlocal equations, Indiana Univ. Math. J., Volume 53 (2004) no. 4, pp. 1095-1126
[11] Funktionalanalysis der Diskretisierungsmethoden, Leipzig, 1976
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