Stability of gravity-capillary solitary waves on shallow water based on the fifth-order Kadomtsev-Petviashvili equation

Longitudinal and transverse instabilities of gravity-capillary solitary waves on shallow water are investigated based on the numerical analysis of the fifth-order Kadomtsev-Petviashvili (KP) equation, which describes the wave phenomena on shallow water where the relevant Bond number is less than and close to 1/3. Two-dimensional (2D) depression gravity-capillary solitary waves are stable to longitudinal perturbations. 2D elevation gravity-capillary solitary waves are unstable to longitudinal perturbations and finally evolve into 2D depression gravity-capillary solitary waves. Three-dimensional (3D) finite-amplitude depression gravity-capillary solitary waves are stable to longitudinal perturbations. 3D finite-amplitude elevation gravity-capillary solitary waves are unstable to longitudinal perturbations and finally evolve into an oscillatory state between two different 3D finite-amplitude depression gravity-capillary solitary waves. 3D small-amplitude depression and elevation gravity-capillary solitary waves are unstable to dilation-type longitudinal perturbations and eventually evolve into an oscillatory state between two different 3D finite-amplitude depression gravity-capillary solitary waves. 3D small-amplitude depression and elevation gravity-capillary solitary waves are unstable to contraction-type longitudinal perturbations and eventually become dispersed out toward still water surface. Finally, 2D depression and elevation gravity-capillary solitary waves are unstable to transverse perturbations and eventually evolve into 3D finite-amplitude depression gravity-capillary solitary waves. Therefore, the only stable gravity-capillary solitary waves on shallow water are 3D finite-amplitude depression gravity-capillary solitary waves. In particular, based on the linear stability analysis, a theoretical proof is presented for the long-wave transverse instability of 2D depression and elevation gravity-capillary solitary waves on shallow water.