Singular travelling wave solutions of the fifth-order KdV, Sawada-Kotera and Kaup equations

Abstract

We obtain second-order equations of degree four (six), for travelling wave solutions of the KdV (Sawada-Kotera/Kaup) equations, which reduce to first-order equations for monotone solitary waves. For the KdV equation, the singular solutions of this equation with an asymptotic value b consist of the well known solution and a new solution with a non-zero asymptotic value depending on the wave speed. We show that the well known solitary wave solutions are determined uniquely as the singular solutions with asymptotic value b = 0, which are also stationary with respect to the wave speed.