A nonlinear Schrödinger equation for capillary waves on arbitrary depth with constant vorticity

A nonlinear Schrödinger equation for pure capillary waves propagating at the free surface of a vertically sheared current has been used to study the stability and bifurcation of capillary Stokes waves on arbitrary depth. A linear stability analysis of weakly nonlinear capillary Stokes waves on arbitrary depth has shown that (i) the growth rate of modulational instability increases as the vorticity decreases whatever the dispersive parameter $kh$ , where $k$ is the carrier wavenumber and $h$ the depth; (ii) the growth rate is significantly amplified for shallow water depths; and (iii) the instability bandwidth widens as the vorticity decreases. Particular attention has been paid to damping due to viscosity and forcing effects on modulational instability. In addition, a linear stability analysis to transverse perturbations in deep water has been carried out, demonstrating that the dominant modulational instability is two-dimensional whatever the vorticity. Near the minimum of linear phase velocity in deep water, we have shown that generalised capillary solitary waves bifurcate from linear capillary Stokes waves when the vorticity is positive. Moreover, we have shown that the envelope of pure capillary waves in deep water is unstable to transverse perturbations. Consequently, deep-water generalised capillary solitary waves are expected to be unstable to transverse perturbations.