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

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-equationut-uxxt+(b+1)uux=buxuxx+uuxxxwith 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.