Two-dimensional solitary water waves with constant vorticity, Part II: the deep capillary case
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910558834393088 |
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| author | Rowan, James Wan, Lizhe |
| author_facet | Rowan, James Wan, Lizhe |
| contents | We consider the two-dimensional capillary water waves with nonzero constant vorticity in infinite depth. We first derive the Babenko equation that describes the profile of the solitary wave. When the velocity $c$ is close to a critical velocity and a sign condition involving the physical parameters is met, the Babenko equation can be reduced to the stationary focusing cubic nonlinear Schrödinger equation plus perturbative error. We show the existence of a critical value of a dimensionless physical parameter below which at least two families of velocities satisfy the focusing condition and above which only one does. This gives the existence of small-amplitude solitary wave solutions for the water wave system with constant vorticity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03428 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two-dimensional solitary water waves with constant vorticity, Part II: the deep capillary case Rowan, James Wan, Lizhe Analysis of PDEs 76B15, 35Q35 We consider the two-dimensional capillary water waves with nonzero constant vorticity in infinite depth. We first derive the Babenko equation that describes the profile of the solitary wave. When the velocity $c$ is close to a critical velocity and a sign condition involving the physical parameters is met, the Babenko equation can be reduced to the stationary focusing cubic nonlinear Schrödinger equation plus perturbative error. We show the existence of a critical value of a dimensionless physical parameter below which at least two families of velocities satisfy the focusing condition and above which only one does. This gives the existence of small-amplitude solitary wave solutions for the water wave system with constant vorticity. |
| title | Two-dimensional solitary water waves with constant vorticity, Part II: the deep capillary case |
| topic | Analysis of PDEs 76B15, 35Q35 |
| url | https://arxiv.org/abs/2408.03428 |