On the spectral stability of periodic capillary-gravity waves

In this paper, we investigate the spectral stability of periodic traveling waves in the two dimensional gravity-capillary water wave problem. We derive a stability criterion based on an index function, whose sign determines the spectral stability of the waves. This result aligns with earlier formal analyses by Djordjevi\'c \& Redekopp [15] and Ablowitz \& Segur [1], which employed the nonlinear Schr\"odinger approximation in the modulational regime. In particular, we show that instability is excluded near spectral crossings away from the origin when the surface tension is positive and the inverse square of the Froude number $\alpha\in(0,1),$ which results from the fact that the corresponding Krein signatures are identical. It is also shown that there exists $\alpha_1 = (23 - 3\sqrt{41})/8$ and a curve $\beta: (\alpha_1, 1]\rightarrow \mathbb{R}_{+},$ such that for any $\alpha \in (\alpha_1, 1]$, small amplitude periodic waves are spectrally stable when $\beta > \beta(\alpha)$. These findings highlight the stabilizing effect of surface tension on periodic capillary-gravity waves.