Second-order Stokes wave solutions for gravity capillary water waves in three-layer dendity-stratified fluid

The gravity-capillary water Waves in two-layer density-stratified fluid are investigated by using a singular perturbation method,and the third-order asymptotic solutions of the velocity potentials and the third-order Stokes wave solutions of the associated elevations of the interfacial waves are presented based on the small amplitude wave theory.As expected,the third-order solutions describe the third-ordei;nonlinear modification and the third-order nonlinear interactions between the interfacial waves.The wave velocity depends on not only the wave number and the depths of each layer,wave amplitude but also the surface tension.