Stability of gravity-capillary solitary waves

Abstract

The stability of gravity-capillary solitary waves on deep water is theoretically and numerically investi-gated using a model equation that admits steep wave patterns near the minimum of the linear gravity-capillary phase speed, $c_min$. Both depression and elevation two-dimensional (plane) gravity-capillary solitary waves of this model are theoretically proven to be unstable to transverse perturbation to the direction of propagation. Indeed, the transverse instability problem of plane gravity-capillary solitary waves on deep water through completely theoretical proof has never been studied

therefore we give a complete theoretical proof of the in-stability of the waves. We also make certain the instability by numerical simulation, which indicates that the instabilities of 2-D elevation and depression waves result in fully localized depression three-dimensional soli-tary waves. The stability of 3-D solitary waves of depression is also concerns, and they are theoretically found to be stable in the presence of longitudinal perturbation. The numerical simulation shows, however, the 3-D solitary waves have more complicated stability characteristics: Small-amplitude waves are unstable, and they may evolve into a finite-amplitude oscillatory state or decay monotonically to zero, while steep waves are stable.