Current modified higher-order Schrödinger equation of broader bandwidth capillary-gravity waves

Abstract

A higher-order nonlinear Schrödinger equation of capillary-gravity waves for broader bandwidth on infinite depth of water including the effect of depth uniform current is established. The derivation is made from Zakharov's integral equation by extending the narrow bandwidth restriction to make it more suited for application to problems involving actual sea waves. On the basis of this equation, a stability analysis is made for uniform Stokes waves. After obtaining an instability condition, instability regions in the perturbed wave number space are displayed that are in good agreement with the exact numerical findings. It is found that the modifications in the stability characteristics at the fourth-order term are due to the interaction between the frequency-dispersion term and the mean flow. It is seen that the growth rate of sideband instability decreases due to the effects of both surface tension and depth uniform following currents. Significant deviations of the instability regions are observed between narrow-banded and broader-banded results. In addition, we have depicted the instability growth rate for the case of pure capillary waves.