| Abstract: |
| Stokes waves are the simplest nontrivial form of water waves, featuring a periodic profile moving steadily in one direction. In the `70s, Benjamin and Feir discovered through experiments that the steady profile is unstable under long-wave perturbations, i.e. disturbances that, although small in amplitude, have much longer period than the initial wave. In recent years, significant mathematical progress has been made on this `modulational` instability, particularly in the linear approximation, which involves understanding the $L^2(\mathbb R)$-spectrum of the water wave operator linearized along a Stokes wave. I will present our latest results on the topic, specifically the full description of arbitrary portions of the unstable spectrum. As long conjectured by numerical investigations, this spectrum consists of infinitely many isolated elliptical branchings, called `isolas`, centered on the imaginary axis that become exponentially small as one moves away from the origin of the complex plane. This work was done in collaboration with M. Berti, L. Corsi, and A. Maspero. |
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