Periodic solutions of the Camassa–Holm equation based on the bilinear form

Abstract

Since the famous work of Camassa and Holm, various analytic techniques have been used to find the exact solutions of Camassa–Holm(CH) equation, and many interesting results have been found by the different methods. In this paper, we first give an extended bilinear form of the CH equation which contains four more free parameters than those in the results proposed in Parkers finding for soliton solutions; thus, we construct the smooth periodic solution of the CH equation in a parametric form based on the previous work of Nakumura. We also show that the periodic solution degenerates into the smooth solitary wave solution by limiting behavior of τ → ∞ (τ is the parameter appeared from the Riemann theta function).