Long-wave transverse instability of weakly nonlinear gravity–capillary solitary waves

Abstract

The long-wave transverse instability of weakly nonlinear gravity–capillary solitary waves is verified from the weakly nonlinear cubic-order truncation model derived from the free-surface boundary conditions of the Euler equations in the water-wave problem. The linearized operators corresponding to the cubic-order truncation model feature a skew-symmetric structure, consistent to the associated property of the Euler equations. From this, the leading-order initial long-wave transverse instability growth rate of the weakly nonlinear gravity–capillary solitary waves is estimated to be quantitatively identical, in the weakly nonlinear limit, to the earlier result obtained from the full Euler equations, through an equivalent perturbation procedure.