Transversal spectral instability of periodic traveling waves for the generalized Zakharov–Kuznetsov equation
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 pos...
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Académie des sciences
2024-07-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.574/ |
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| author | Natali, Fábio |
| author_facet | Natali, Fábio |
| author_sort | Natali, Fábio |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-adc0b363d86a4450971df9310085fd3d |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-07-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-adc0b363d86a4450971df9310085fd3d2025-02-07T11:21:51ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-07-01362G660761710.5802/crmath.57410.5802/crmath.574Transversal spectral instability of periodic traveling waves for the generalized Zakharov–Kuznetsov equationNatali, Fábio0Departamento de Matemática – Universidade Estadual de Maringá Avenida Colombo, 5790, CEP 87020-900, Maringá, PR, BrazilIn 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.574/ |
| spellingShingle | Natali, Fábio Transversal spectral instability of periodic traveling waves for the generalized Zakharov–Kuznetsov equation Comptes Rendus. Mathématique |
| title | Transversal spectral instability of periodic traveling waves for the generalized Zakharov–Kuznetsov equation |
| title_full | Transversal spectral instability of periodic traveling waves for the generalized Zakharov–Kuznetsov equation |
| title_fullStr | Transversal spectral instability of periodic traveling waves for the generalized Zakharov–Kuznetsov equation |
| title_full_unstemmed | Transversal spectral instability of periodic traveling waves for the generalized Zakharov–Kuznetsov equation |
| title_short | Transversal spectral instability of periodic traveling waves for the generalized Zakharov–Kuznetsov equation |
| title_sort | transversal spectral instability of periodic traveling waves for the generalized zakharov kuznetsov equation |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.574/ |
| work_keys_str_mv | AT natalifabio transversalspectralinstabilityofperiodictravelingwavesforthegeneralizedzakharovkuznetsovequation |