The mathematical study of rogue wave phenomenon had been on-going for years. Rogue waves are unusually large-amplitude surface waves that appear from nowhere in the open ocean. Collisions with such waves have caused catastrophic damage to ships and offshore structures. In this work, we apply an analytic technique, namely the Homotopy Analysis Method (HAM) to derive the rogue wave solution for the fully nonlinear wave equations with nonlinear boundary conditions. On the basis of the HAM, we obtain an analytic expression for the surface wave elevation η and velocity potential for a rogue wave. The expressions obtained for η are exact and depends on the computed values of the velocity potential. Due to the highly nonlinear nature of the problem (having nonlinear boundary conditions) the velocity potentials are obtained up to 5th order approximation. Surface plots of the wave profile are presented. These show high level agreement with the famous New Year wave at the Draupner platform. It is expected that this study will deepen and enrich our understanding of rogue waves.
Keywords: homotopy analysis method, rogue wave, surface wave elevation, velocity potential
