Scattering for the Quartic Generalized Benjamin-Bona-Mahony Equation
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929245322739712 |
|---|---|
| author | Morgan, A. George |
| author_facet | Morgan, A. George |
| contents | The generalized Benjamin-Bona-Mahony equation (gBBM) is a model for nonlinear dispersive waves which, in the long-wave limit, is approximately equivalent to the generalized Korteweg-de Vries equation (gKdV). While the long-time behaviour of small solutions to gKdV is well-understood, the corresponding theory for gBBM has progressed little since the 1990s. Using a space-time resonance approach, I establish linear dispersive decay and scattering for small solutions to the quartic-nonlinear gBBM. To my knowledge, this result provides the first global-in-time pointwise estimates on small solutions to gBBM with a nonlinear power less than or equal to five. Owing to nonzero inflection points in the linearized gBBM dispersion relation, there exist isolated space-time resonances without null structure, but in the course of the proof I show these resonances do not obstruct scattering. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09605 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Scattering for the Quartic Generalized Benjamin-Bona-Mahony Equation Morgan, A. George Analysis of PDEs 35Q35, 76B15 The generalized Benjamin-Bona-Mahony equation (gBBM) is a model for nonlinear dispersive waves which, in the long-wave limit, is approximately equivalent to the generalized Korteweg-de Vries equation (gKdV). While the long-time behaviour of small solutions to gKdV is well-understood, the corresponding theory for gBBM has progressed little since the 1990s. Using a space-time resonance approach, I establish linear dispersive decay and scattering for small solutions to the quartic-nonlinear gBBM. To my knowledge, this result provides the first global-in-time pointwise estimates on small solutions to gBBM with a nonlinear power less than or equal to five. Owing to nonzero inflection points in the linearized gBBM dispersion relation, there exist isolated space-time resonances without null structure, but in the course of the proof I show these resonances do not obstruct scattering. |
| title | Scattering for the Quartic Generalized Benjamin-Bona-Mahony Equation |
| topic | Analysis of PDEs 35Q35, 76B15 |
| url | https://arxiv.org/abs/2402.09605 |