  The well-known Stokes waves refer to periodic traveling waves under the
gravity at the free surface of a two dimensional full water wave system. In
this paper, we prove that small-amplitude Stokes waves with infinite depth are
nonlinearly unstable under long-wave perturbations. Our approach is based on
the modulational approximation of the water wave system and the instability
mechanism of the focusing cubic nonlinear Schr\"odinger equation.
