| Abstract: |
| We study the modulational (Benjamin-Feir) instability of small-amplitude periodic Stokes waves for the two-dimensional gravity-capillary water waves equations. We analyze the spectral stability of such waves under long-wave longitudinal perturbations.
The linearized operator exhibits a defective zero eigenvalue of multiplicity four due to the symmetries of the system. Using Bloch-Floquet theory, we investigate the associated family of periodic spectral problems and obtain a complete description of the eigenvalues near the origin in the small-amplitude regime.
We prove that the four eigenvalues undergo a full splitting and rigorously characterize the transition between stability and instability. In the unstable regime, the spectrum near the origin forms the characteristic Figure 8 pattern associated with Benjamin-Feir instability, while in the stable regime it remains purely imaginary. |
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