On the short-wavelength stability and the Kelvin–Helmholtz instability of some explicit solutions to the two-layer nonlinear three-dimensional water wave problem

This study is concerned with explicit solutions in the Eulerian framework to the three-dimensional nonlinear Euler equations with a free surface and an interface. Starting with a depth-dependent density, continuous within the fluid domain, except across the interface, we present exact radial solutions to the steady water wave problem. While the velocity field and the pressure are given explicitly, the free surface and the interface are determined implicitly by a functional analytic approach. We also prove short-wavelength stability results for these explicit solutions by means of a Wentzel–Kramers–Brillouin ansatz together with a Lyapunov-type stability criterion. We conclude the analysis by proving the Kelvin–Helmholtz instability of the exact solutions.