On the spectra of the gravity water waves linearized at monotone shear flows

We consider the spectra of the 2-dim gravity waves of finite depth linearized at a uniform monotonic shear flow $U(x_2)$, $x_2\in (-h,0)$, where the wave numbers $k$ of the horizontal variable $x_1$ is treated as a parameter. Our main results include a.) a complete branch of non-singular neutral modes $c^{+}(k)$ strictly decreasing in $k\ge 0$ and converging to $U(0)$ as $k\to \mathrm{\infty }$; b.) another branch of non-singular neutral modes $c_{-}(k)$, $k\in (-k_{-},k_{-})$ for some $k_{-}>0$, with $c_{-}(\pm k_{-})=U(-h)$; c.) the non-degeneracy and the bifurcation at $(k_{-},c=U(-h))$; d.) the existence and non-existence of unstable modes for $c$ near $U(0)$, $U(-h)$, and interior inflection values of $U$; e.) the complete spectral distribution in the case where $U^{\mathrm{\prime }\mathrm{\prime }}$ does not change sign or changes sign exactly once non-degenerately. In particular, $U$ is spectrally stable if $U^{\mathrm{\prime }}{U}^{\mathrm{\prime }\mathrm{\prime }}\le 0$ and unstable if $U$ has a non-degenerate interior inflection value or $\{U^{\mathrm{\prime }}{U}^{\mathrm{\prime }\mathrm{\prime }}>0\}$ accumulate at $x_2=-h$ or $0$. Moreover, if $U$ is an unstable shear flow of the fixed boundary problem in a channel, then strong gravity could cause instability of the linearized gravity waves in all long waves (i.e. $\mathrm{\mid }k\mathrm{\mid }\ll 1$).