Periodic cusp wave solutions and single-solitons for the b-equation

Abstract: In this paper, we employ the bifurcation method of dynamical systems and the numerical simulation approach of differential equations to study periodic cusp wave solutions and single-solitons for the b-equationwith b>1. The explicit representations of periodic cusp waves and the implicit expressions of single-solitons are obtained. Further, we show that the limits of both periodic cusp waves and single-solitons are peakons which possess explicit expression u=ce−∣x−ct∣. As corollary, the single-solitons equations of the Camassa–Holm equation and the Degasperis–Procesi equation are given. Our theoretical derivations are identical with the numerical simulations. [Copyright &y& Elsevier]