PERIODIC WAVES AND THEIR LIMITS FOR THE CAMASSA–HOLM EQUATION

Abstract

In this paper, the bifurcation method of dynamical systems is employed to study the Camassa–Holm equation [Formula: see text] We investigate the periodic wave solutions of form u = φ(ξ) which satisfy φ(ξ + T) = φ(ξ), here ξ = x - ct and c, T are constants. Their six implicit expressions and two explicit expressions are obtained. We point out that when the initial values are changed, the periodic waves may become three waves, periodic cusp waves, smooth solitary waves and peakons. Our results give an explanation to the appearance of periodic cusp waves and peakons. Moreover, three sets of graphs of the implicit functions are drawn, and three sets of numerical simulations are displayed. The identity of these graphs and simulations imply the correctness of our theoretical results.