From KP-I Lump Solution to Travelling wave of 3D Gravity Capillary Water wave problem

Abstract: In this paper, we study the three-dimensional gravity-capillary water wave problem involving an irrotational, perfect fluid with gravity and surface tension. We focus on steady waves propagating uniformly in one direction. Assuming constant wave speed and water depth, we analyze the fluid's velocity potential and boundary conditions. Using the Kadomtsev-Petviashvili (KP)-I equation as a simplified model, we show that, within a specific parameter range, the problem admits a fully localized solitary-wave solution resemble lump type solutions of the KP-I equation.