Stability of periodic gravity waves in the presence of surface tension

The goal of this work is to investigate the effect of the inclusion of small surface tension on the instabilities of periodic gravity water waves that are present even in shallow water (Deconinck and Oliveras, 2011). Using the recent reformulation of Ablowitz, Fokas and Musslimani (2006), we compute periodic traveling water waves where the effects of both gravity and small surface tension are incorporated. The spectral stability of these solutions is examined using Hill's method (Deconinck and Kutz, 2006). It is found that the instabilities are not suppressed by the inclusion of surface tension. In fact, the growth rates associated with them increase as the surface tension grows. Generalizing the work of MacKay and Saffman (1986), the persistence of these instabilities is confirmed analytically for waves of small amplitude.