On the Camassa-Holm equation and a direct method of solution I. Bilinear form and solitary waves

Abstract

Over the last decade, a variety of techniques has been used to find exact solutions (both analytic and other) of the Camassa–Holm equation. The different approaches have met with varying measures of success in eliciting the important class of soliton solutions. In this, the first of two papers, we show how Hirota's bilinear transform method can be used to obtain analytic solutions of the Camassa–Holm equation. A bilinear form of the Camassa–Holm equation is presented and used to derive the solitary–wave solution, which is examined in various parameter regimes. A limiting procedure is then used to recover the well–known non–analytic ‘peakon’ solution from the solitary wave. The results reported here provide a basis for constructing explicitly the erstwhile elusive N–soliton solutions of the Camassa–Holm equation in a sequel paper.