Nonlinear Interaction between Short and Long Capillary-Gravity Waves

Coupled nonlinear partial differential equations, which describe a nonlinear interaction between short and long capillary-gravity waves on a liquid layer of uniform depth, are derived by the derivative expansion method. The short and the long waves can exchange energy in a resonant manner, if the group velocity of the short wave is close to the phase velocity of the long wave. It is found that the long wave can take a form of rarefactive (convex downwards) solitary wave due to the resonant interaction. This should be compared with the well-known gravity wave soliton which is compressive (convex upwards) in the absence of the short wave.