Spectral structure of the Benjamin-Feir instability in deep-water gravity-capillary Stokes waves
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910142872682496 |
|---|---|
| author | Hsiao, Ting-Yang Wang, Xinyang |
| author_facet | Hsiao, Ting-Yang Wang, Xinyang |
| contents | We investigate the Benjamin-Feir instability of small-amplitude gravity-capillary Stokes waves in deep water for the full water wave equations. While modulational instability has been classically predicted by formal asymptotic approaches, such as nonlinear Schrödinger approximations, a complete spectral description at the level of the Euler equations has remained open. We perform a rigorous Bloch-Floquet spectral analysis of the linearized operator and describe the splitting of the multiple eigenvalues at the origin. In the unstable regime, we identify a pair of eigenvalues with non-zero real part forming the characteristic ``figure-eight'' pattern in the complex plane. As a consequence, we recover sharp instability and stability regions in terms of the surface tension parameter, thereby providing a fully rigorous justification of the classical predictions in the gravity-capillary setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_16724 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral structure of the Benjamin-Feir instability in deep-water gravity-capillary Stokes waves Hsiao, Ting-Yang Wang, Xinyang Analysis of PDEs 76B15, 35P05, 35B35, 35B10 We investigate the Benjamin-Feir instability of small-amplitude gravity-capillary Stokes waves in deep water for the full water wave equations. While modulational instability has been classically predicted by formal asymptotic approaches, such as nonlinear Schrödinger approximations, a complete spectral description at the level of the Euler equations has remained open. We perform a rigorous Bloch-Floquet spectral analysis of the linearized operator and describe the splitting of the multiple eigenvalues at the origin. In the unstable regime, we identify a pair of eigenvalues with non-zero real part forming the characteristic ``figure-eight'' pattern in the complex plane. As a consequence, we recover sharp instability and stability regions in terms of the surface tension parameter, thereby providing a fully rigorous justification of the classical predictions in the gravity-capillary setting. |
| title | Spectral structure of the Benjamin-Feir instability in deep-water gravity-capillary Stokes waves |
| topic | Analysis of PDEs 76B15, 35P05, 35B35, 35B10 |
| url | https://arxiv.org/abs/2604.16724 |