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

Nonlinear modulational instabililty of the Stokes waves in 2d full water waves