Abstract
The interaction among a quartet of resonant progressive waves in water of finite depth is considered by the Homotopy Analysis Method (HAM). The problem is governed by a linear PDE with a group of nonlinear boundary conditions on the unknown free surface. Convergent multiple steady-state solutions have been gained by means of the HAM. It's worth noting that, in the most cases of the quartet resonance, wave energy is often exchanged periodically among different wave modes. However, our computations indicate for the first time that there exist some cases in which energy exchange disappears and besides the resonant wave component may contain much small percentage of the total wave energy. This work verifies that the HAM is valid even for some rather complicated nonlinear PDEs, and can be used as a powerful tool to understand some nonlinear phenomena.