Transverse instability of gravity-capillary line solitary water waves

The gravity-capillary water-wave problem concerns the irrotational flow of a perfect fluid in a domain bounded below by a rigid bottom and above by a free surface under the influence of gravity and surface tension. In the case of large surface tension the system has a travelling line solitary-wave solution for which the free surface has a localised profile in the direction of propagation and is homogeneous in the transverse direction. In this note we show that this line solitary wave is linearly unstable under spatially inhomogeneous perturbations which are periodic in the direction transverse to propagation.