The homogeneous balance method, Lax pair, Hirota transformation and a general fifth-order KdV equation

In this paper, some notes of the homogeneous balance (HB) method are discussed by a kind of general fifth-order KdV (fKdV) equation. First, the auto-Bäcklund transformation and Lax represents of the higher-order KdV equation (a specific form of fKdV equation) are obtained by the HB method. Then, the connection of the Hirota transformation and the HB method is discussed by two specific fKdV equations: the Sawada–Kotera equation and the Laxʹs equation. At the same time, the solitary wave solution of the general fKdV equation is obtained.