On soliton solutions of the Kaup–Kupershmidt equation. II. ‘Anomalous’ N-soliton solutions

In the second of two articles (designated I [Physica D 137 (2000) 25] and II), exact soliton solutions of the Kaup–Kupershmidt equation are derived using a simplified approach based in part on Hirota’s bilinear transform. These solutions have not previously been obtained explicitly by analytic methods. It is found that, at each order, the N-soliton solution is characterised by an additional parameter which accounts for the ‘anomalous’ nature of these solutions. The first four solitons are constructed explicitly, and in a novel departure from Hirota’s method, an iterative procedure is presented for obtaining the general N-soliton solution. Some possible alternative approaches to solving the Kaup–Kupershmidt equation are considered.