A note on exact travelling wave solutions for the modified Camassa–Holm and Degasperis–Procesi equations

Abstract

Exact solutions for the modified Camassa–Holm and Degasperis–Procesi equations by Liu et al. (2010) [Y.F. Liu, X.Y. Zhu, J.X. He, Factorization technique and new exact solutions for the modified Camassa–Holm and Degasperis–Procesi equations, Appl. Math. Comput. 217 (2010) 1658–1665] are investigated. Liu et al. has used the factorization technique to reduce the modified Camassa–Holm and Degasperis–Procesi equations to first-order ordinary differential equations, and then derived some exact travelling wave solutions by direct integral method. In this note, we will explain that the implementation of the so-called factorization technique is completely unnecessary. Moreover, based on the method of complete discrimination system for polynomial, we shall demonstrate that the general explicit exact solution and its classification for the above two types of equations can be obtained directly and many exact solutions by Liu et al. are our special cases. Besides, some known results in previously relevant literatures are extended and some simple remarks are also made.