On weakly nonlinear gravity-capillary solitary waves

As a weakly nonlinear model equations system for gravity-capillary solitary waves on the surface of a potential flow, a cubic-order truncation model is presented, which is derived from the Taylor series expansion of the Dirichlet-Neumann operator (DNO) for the free boundary conditions of the Euler equations in terms of Zakharov's canonical variables. In deep water, the cubic-order truncation model allows gravity-capillary solitary wavepackets in the weakly nonlinear and narrow bandwidth regime where the classical nonlinear Schr\"{o}dinger (NLS) equation governs. Since this model is consistent to the original full Euler equations in the order of nonlinearity up to the third order, the properties of the gravity-capillary solitary waves of this model precisely agree with the counterparts of the Euler equations. From this cubic order truncation model, the leading-order initial long-wave transverse instability growth rate of the gravity-capillary solitary waves is estimated to be identical, in the weakly nonlinear limit, to the earlier result by Kim and Akylas (J. Eng. Math. 58:167-175, 2007), through an equivalent perturbation procedure. Based on these analytical and numerical observations, the cubic-order truncation model equations system is regarded as the optimal reduced model for the dynamics of weakly nonlinear gravity-capillary solitary waves.