Comptes Rendus
no. 13-14
Partial Differential Equations/Numerical Analysis
On a nonlocal moving frame approximation of traveling waves
[Approximation dʼondes progressives dans un référentiel non-local en mouvement]

Présenté par : Roland Glowinski

Jose M. Arrieta1 ; Maria López-Fernández2 ; Enrique Zuazua3, 4
1 Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
2 Institut für Mathematik, Universität Zürich, Winterthurerst, 190, CH-8057 Zurich, Switzerland
3 Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, 48011, Bilbao, Basque Country, Spain
4 BCAM – Basque Center for Applied Mathematics, Bizkaia Technology Park 500, 48160, Derio, Basque Country, Spain
Comptes Rendus. Mathématique, Volume 349 (2011) no. 13-14, pp. 753-758.

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 R. This leads to efficient numerical algorithms for computing the traveling wave profile and speed of propagation.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.07.001
Jose M. Arrieta 1 ; Maria López-Fernández 2 ; Enrique Zuazua 3, 4

1 Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
2 Institut für Mathematik, Universität Zürich, Winterthurerst, 190, CH-8057 Zurich, Switzerland
3 Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, 48011, Bilbao, Basque Country, Spain
4 BCAM – Basque Center for Applied Mathematics, Bizkaia Technology Park 500, 48160, Derio, Basque Country, Spain 
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     author = {Jose M. Arrieta and Maria L\'opez-Fern\'andez and Enrique Zuazua},
     title = {On a nonlocal moving frame approximation of traveling waves},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {753--758},
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     volume = {349},
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     doi = {10.1016/j.crma.2011.07.001},
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Jose M. Arrieta; Maria López-Fernández; Enrique Zuazua. On a nonlocal moving frame approximation of traveling waves. Comptes Rendus. Mathématique, Volume 349 (2011) no. 13-14, pp. 753-758. doi : 10.1016/j.crma.2011.07.001. https://comptes-rendus.academie-sciences.fr/mathematique/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] W.J. Beyn; J. Rottman-Matthes 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] W.J. Beyn; V. Thümmler Freezing solutions of equivariant evolution equations, SIAM J. Appl. Dyn. Syst., Volume 3 (2004), pp. 85-116

[4] W.J. Beyn; V. Thümmler Phase conditions, symmetries, and PDE continuation, Numerical continuation methods for dynamical systems, Underst. Complex Syst., Springer, Dordrecht, 2007, pp. 301-330

[5] F.A. Davidson; N. Dodds Spectral properties of non-local uniformly-elliptic operators, Electron. J. Differential Equations, Volume 126 (2006), p. 15

[6] N. Dodds Further spectral properties of uniformly elliptic operators that include a non-local term, Appl. Math. Comput., Volume 197 (2008), pp. 317-327

[7] P.C. Fife; J.B. McLeod The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rational Mech. Anal., Volume 65 (1977), pp. 335-361

[8] P. Freitas Non-local reaction diffusion equations, Fields Institute Communications, Volume 21 (1999), pp. 187-204

[9] D. Henry Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer, Berlin, 1981

[10] T. Kapitula; N. Kutz; B. Sandstede The Evans function for nonlocal equations, Indiana Univ. Math. J., Volume 53 (2004) no. 4, pp. 1095-1126

[11] G. Vainikko Funktionalanalysis der Diskretisierungsmethoden, Leipzig, 1976

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