Comptes Rendus
Research article - Partial differential equations
Transversal spectral instability of periodic traveling waves for the generalized Zakharov–Kuznetsov equation
Fábio Natali1
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.

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.

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.

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Accepted:
Published online:
DOI: 10.5802/crmath.574
Classification: 35B10, 35B35, 35Q53

Fábio Natali 1

1 Departamento de Matemática – Universidade Estadual de Maringá Avenida Colombo, 5790, CEP 87020-900, Maringá, PR, Brazil
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     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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