On soliton solutions of the Kaup–Kupershmidt equation. I. Direct bilinearisation and solitary wave

In this first of two papers (designated I and II [Physica D 137 (2000) 34]), a direct procedure is given for bilinearising the Kaup–Kupershmidt and Sawada–Kotera equations. It is found that, whereas the Sawada–Kotera equation can be written in bilinear form in two distinct ways (which are shown to be equivalent), the Kaup–Kupershmidt equation has a single bilinear representation. This distinguishing feature, which has previously received no explanation, has consequences for obtaining the soliton solutions of these two closely related fifth-order integrable equations. The ‘anomalous’ solitary wave of the Kaup–Kupershmidt equation is reconstructed, and a simplified approach to obtaining its N-soliton solutions is proposed. This provides the basis for constructing these solutions explicitly in II.