For a partial differential equation in spatial dimension one, admitting a spatially homogeneous time periodic solution, we show the generic existence, close to this solution, of a one-parameter family of travelling waves parametrized by their wave number k (k=0 corresponding to the spatially homogeneous initial solution). The argument is elementary and relies on a direct application of singular perturbation theory (Fenichel's global center manifold theorem).
Pour une équation aux dérivées partielles en dimension un d'espace, admettant une solution homogène en espace et périodique en temps, on montre l'existence, au voisinage de cette solution, d'une famille à un paramètre d'ondes progressives paramétrisées par leur nombre d'onde k (k=0 correspondant à la solution spatialement homogène initiale). La justification, élémentaire, est basée sur un argument de perturbation singulière (théorème de la variété centrale globale de Fenichel).
Accepted:
Published online:
Emmanuel Risler 1
@article{CRMATH_2002__334_9_833_0, author = {Emmanuel Risler}, title = {Travelling waves and dispersion relation in the spatial unfolding of a periodic orbit}, journal = {Comptes Rendus. Math\'ematique}, pages = {833--838}, publisher = {Elsevier}, volume = {334}, number = {9}, year = {2002}, doi = {10.1016/S1631-073X(02)02363-4}, language = {en}, }
Emmanuel Risler. Travelling waves and dispersion relation in the spatial unfolding of a periodic orbit. Comptes Rendus. Mathématique, Volume 334 (2002) no. 9, pp. 833-838. doi : 10.1016/S1631-073X(02)02363-4. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02363-4/
[1] On limit cycles in systems of differential equations with a small parameter in the highest derivatives, Amer. Math. Soc. Transl. Sér. 2, Volume 33 (1963), pp. 233-275 Translation from Mat. Sb. (N.S.) 50 (92) (1960) 299–334
[2] Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, Volume 31 (1979), pp. 53-98
[3] Geometric singular perturbation theory, Lecture Notes in Math., 1609, 1995, pp. 44-118
Cited by Sources:
Comments - Policy