The Fourth-Order Nonlinear Schrödinger Equation and Stability Analysis for Stokes Waves on Slowly Varying Topography

Abstract

The surface gravity wave equation is expanded to the fourth-order wave steepness on slowly varying topography, obtaining a topographic modified nonlinear Schrödinger (TMNLS) equation. When the time scale is longer than ε-3 times of the dominant wave period or the space scale is larger than ε-3 times the dominant wavelength, the second water depth derivative and the square of the first water depth derivative affect the first-order wave amplitude. The instability area for a uniform Stokes wave train by small perturbations is the entire wavenumber space, except for a specific stability curve on infinite and slowly varying depth. The depth variation terms affect the growth rate of uniform Stokes wave train on the order of 0.01. The stability curve shows more sensitive to the depth variation in x direction than that in y direction. The increment of the value for depth variation in x direction contributes the stable wave number of perturbation to approach or parallel to y axis. The increment of the value for depth variation in y direction helps the stable wave number of perturbation to approach or parallel to x axis.