Transversal spectral instability of periodic traveling waves for the generalized Zakharov–Kuznetsov equation

Fábio Natali

doi:10.5802/crmath.574

Article de recherche - Equations aux dérivées partielles

Transversal spectral instability of periodic traveling waves for the generalized Zakharov–Kuznetsov equation
[Instabilité spectrale transversale des ondes de déplacement périodiques pour l’équation de Zakharov–Kuznetsov généralisée]

Fábio Natali ¹

¹ Departamento de Matemática – Universidade Estadual de Maringá Avenida Colombo, 5790, CEP 87020-900, Maringá, PR, Brazil

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 607-617.

Résumé
Abstract

Dans cet article, nous déterminons l’instabilité transversale des solutions périodiques de l’équation de Zakharov–Kuznetsov généralisée en deux dimensions spatiales. En utilisant l’adaptation des arguments de [F. Rousset et N. Tzvetkov, 2010] dans le contexte périodique, il est possible de prouver que toutes les ondes positives et unidimensionnelles L’ sont spectralement (transversalement) instables. En outre, lorsqu’il existe des ondes périodiques qui changent de signe, nous obtenons également la même propriété lorsque l’opérateur de projection associé défini dans l’espace de Sobolev à moyenne nulle n’a qu’une seule valeur propre négative.

In this paper, we determine the transversal instability of periodic traveling wave solutions of the generalized Zakharov–Kuznetsov equation in two space dimensions. Using an adaptation of the arguments in [F. Rousset et N. Tzvetkov, 2010] in the periodic context, it is possible to prove that all positive and one-dimensional $L -$ periodic waves are spectrally (transversally) unstable. In addition, when periodic waves that change their sign exist, we also obtain the same property when the associated projection operator defined in the zero mean Sobolev space has only one negative eigenvalue.

Reçu le : 2023-03-21
Révisé le : 2023-09-14
Accepté le : 2023-10-04
Publié le : 2024-07-09

DOI : 10.5802/crmath.574

Classification : 35B10, 35B35, 35Q53

Affiliations des auteurs :

Fábio Natali ¹

¹ Departamento de Matemática – Universidade Estadual de Maringá Avenida Colombo, 5790, CEP 87020-900, Maringá, PR, Brazil

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2024__362_G6_607_0,
     author = {F\'abio Natali},
     title = {Transversal spectral instability of periodic traveling waves for the generalized {Zakharov{\textendash}Kuznetsov} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {607--617},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.574},
     language = {en},
}

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Fábio Natali. Transversal spectral instability of periodic traveling waves for the generalized Zakharov–Kuznetsov equation. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 607-617. doi : 10.5802/crmath.574. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.574/

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