  We generalize the periodic Evans function approach recently used to study the
spectral stability of Stokes wave and gravity-capillary (including Wilton
ripples) in water of finite depth to study spectral stability of Stokes waves
in water of infinite depth. We prove waves of sufficiently small amplitude are
always low-frequency unstable regardless of the wave number and gravity, giving
an alternative proof for the Benjamin-Feir modulational instability in the
infinite depth case. Here, the first proof for the infinite depth case is
recently obtained by Nguyen and Strauss. We also study the spectral stability
at non-zero resonant frequencies and find no additional instability.

Analysis of PDEs