Mathematics

Traveling water waves are susceptible to high frequency instabilities, i.e., instabilities that are different from the modulational or Benjamin-Feir instability: the spectrum of the linearized problem contains isole of unstable eigenvalues away from the origin of the spectral plane. These instabilities have been observed numerically, and recently different researchers have been trying to approach them through analytical means. I will present a perturbation scheme that allows us to provide highly accurate approximations to the numerically
observed isole, for the water wave problem (in progress) and for simpler problems where such instabilities are present as well. To our knowledge, this presents the first quantitative comparison of such results with numerics.