Some Explicit Solutions for Two-Layer Three-Dimensional Gravity Water Waves

We present explicit solutions for two-layer gravity water waves with depth-dependent density, analyzed within the Eulerian framework. Both time-independent and time-dependent solutions are considered. For the former case, we construct a family of solutions with radial structures. For the latter case, under the piecewise constant densities, we derive a class of explicit solutions where the time-dependent term must be (and can only be) a polynomial. For the depth-dependent density, we construct a new class of exact solutions with more complex time-dependent terms, and provide several representative examples. This type of solution has not been presented elsewhere before. Furthermore, the results we obtained can be extended to n-layer fluids.