TWO-DIMENSIONAL SOLITARY WAVE SOLUTIONS FOR WATER WAVES NEAR THE CRITICAL POINT

This paper considers that the Bond number is greater than but close to $\frac{1}{3} $ and the Froude number is greater than but close to 1 for the two-dimensional traveling gravity-capillary waves in water of finite depth. The horizontal propagating direction and the stream function are chosen as independent variables. The Euler equations are reduced to a system of ordinary differential equations with dimension 4 by applying a spatial dynamics approach, a center manifold reduction and a normal form analysis. The reduced system has a homoclinic solution near a nonzero equilibrium. The fixed point theorem shows that this homoclinic solution persists for the original system, i.e., the hydrodynamic problem has a solitary wave solution which exponentially approaches a constant function (independent of the horizontal variable but dependent on the stream function) at infinity.