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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.16772 |
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| _version_ | 1866918030899937280 |
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| author | Pei, Long Xiao, Fengyang Zhang, Pan |
| author_facet | Pei, Long Xiao, Fengyang Zhang, Pan |
| contents | We consider the traveling structure of symmetric solutions to the Rosenau-Kawahara-RLW equation and the perturbed R-KdV-RLW equation. Both equations are higher order perturbations of the classical KdV equation. For the Rosenau-Kawahara-RLW equation, we prove that classical and weak solutions with a priori symmetry must be traveling solutions. For the more complicated perturbed R-KdV-RLW equation, we classify all symmetric traveling solutions, and prove that there exists no nontrivial symmetric traveling solution of solitary type once dissipation or shoaling perturbations exist. This gives a new perspective for evaluating the suitableness of a model for water waves. In addition, this result illustrates the sharpness of the symmetry principle in [Int. Math. Res. Not. IMRN, 2009; Ehrnstrom, Holden \& Raynaud] for solitary waves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16772 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the steadiness of symmetric solutions to higher order perturbations of KdV Pei, Long Xiao, Fengyang Zhang, Pan Analysis of PDEs We consider the traveling structure of symmetric solutions to the Rosenau-Kawahara-RLW equation and the perturbed R-KdV-RLW equation. Both equations are higher order perturbations of the classical KdV equation. For the Rosenau-Kawahara-RLW equation, we prove that classical and weak solutions with a priori symmetry must be traveling solutions. For the more complicated perturbed R-KdV-RLW equation, we classify all symmetric traveling solutions, and prove that there exists no nontrivial symmetric traveling solution of solitary type once dissipation or shoaling perturbations exist. This gives a new perspective for evaluating the suitableness of a model for water waves. In addition, this result illustrates the sharpness of the symmetry principle in [Int. Math. Res. Not. IMRN, 2009; Ehrnstrom, Holden \& Raynaud] for solitary waves. |
| title | On the steadiness of symmetric solutions to higher order perturbations of KdV |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.16772 |