On solitary-wave solutions of fifth-order KdV type of model equations for water waves

Abstract

<p style='text-indent:20px;'>The paper concerns capillary-gravity surface waves propagating on a two-dimensional incompressible and inviscid fluid with or without an obstacle placed at a flat horizontal bottom. Formally second-order correct model equations, analogous to the first-order correct KdV equation, are derived under the classical assumptions of long wavelength and small amplitude of the surface waves. The model equations are a fifth-order KdV type of equations where the obstacle at the bottom introduces forcing terms in the equations. If the forcing is zero, the equation can possess a special Hamiltonian structure after a BBM term is introduced. The existence and set stability of solitary-wave solutions for the fifth-order KdV-BBM type of equations without forcing are studied. If the forcing term is included, only a special case is considered, which is the forced fifth-order KdV type equation (also called forced Kawahara equation) and models the water waves when both the surface tension and wave speed are near their critical values. The existence of small or some large solitary-wave solutions is obtained and the Lyapunov stability of small solitary-wave solutions is proved. Other large amplitude solutions are studied numerically.</p>