Linear-shear-current modified nonlinear Schrödinger equation for gravity-capillary waves on deep water

Starting from Zakharov’s integral equation (ZIE) a modified nonlinear Schrödinger equation (NLSE) correct to fourth-order in wave steepness for deep water gravity-capillary waves (GCW) on linear shear currents (LSC) is derived under the assumption of narrow bandwidth. This equation is then used to examine the stability of uniform wave train. It is found that LSC change considerably the instability behaviors of weakly nonlinear GCW. At both third and fourth-orders, we have shown the significance of nonlinear coupling between the wave-induced mean flow and the vorticity. The key result is that the new fourth-order analysis shows notable deviations in the modulational instability properties from the third-order analysis and provides better results consistent with the exact results. The united effect of vorticity and surface tension is to increase the modulational growth rate of instability influenced by surface tension when the vorticity is negative. As it turns out, the most significant contribution appears from the mean flow response and in the absence of vorticity and depth uniform current the effect of mean flow for pure capillary waves is of opposite sign to that of pure gravity waves. As a consequence, it modifies significantly the modulational instability properties.