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.