Scattering for the quintic generalized Benjamin-Bona-Mahony equation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911478246801408 |
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| author | Chen, Gong Zhang, Yingmo |
| author_facet | Chen, Gong Zhang, Yingmo |
| contents | We consider the quintic generalized Benjamin-Bona-Mahony equation $$
u_t-u_{xxt} + \partial_x\big(u + u^{5}\big)= 0,\qquad (t,x)\in \mathbb{R}_+ \times \mathbb{R}. $$ Using the space-time resonance method, we prove that sufficiently small and smooth solutions scatter to the linear flow.
While the higher nonlinearity simplifies the treatment of nonresonant interactions compared to the quartic model in \cite{Morgan}, resonance analysis is more intricate. The resonance analysis occurs in a higher-dimensional geometric setting, and certain null or vanishing conditions present in the quartic case fail at specific resonance points. As a result, refined computations and precise estimates near the resonant set are required to close the bootstrap argument. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_01360 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Scattering for the quintic generalized Benjamin-Bona-Mahony equation Chen, Gong Zhang, Yingmo Analysis of PDEs 76B15 We consider the quintic generalized Benjamin-Bona-Mahony equation $$ u_t-u_{xxt} + \partial_x\big(u + u^{5}\big)= 0,\qquad (t,x)\in \mathbb{R}_+ \times \mathbb{R}. $$ Using the space-time resonance method, we prove that sufficiently small and smooth solutions scatter to the linear flow. While the higher nonlinearity simplifies the treatment of nonresonant interactions compared to the quartic model in \cite{Morgan}, resonance analysis is more intricate. The resonance analysis occurs in a higher-dimensional geometric setting, and certain null or vanishing conditions present in the quartic case fail at specific resonance points. As a result, refined computations and precise estimates near the resonant set are required to close the bootstrap argument. |
| title | Scattering for the quintic generalized Benjamin-Bona-Mahony equation |
| topic | Analysis of PDEs 76B15 |
| url | https://arxiv.org/abs/2603.01360 |