Abstract: I will discuss spectral instability of a Stokes wave of small amplitude in the finite depth. Analysis of the periodic Evans function near the origin of the spectral plane offers an alternative proof of the Benjamin-Feir instability. Analysis near a pair of resonance frequencies reveals spectral instability when 0.8644...<(wave number)x(depth)<1.0079.... The Benjamin-Feir instability occurs when (wave number)x(depth)>1.3627..., so new unstable waves are found. This seems the first rigorous proof of the high-frequency instability. Joint work with Z. Yang.
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