New compact and noncompact solutions for two variants of a modified Camassa-Holm equation

In this paper exact solutions of two variants of a modified Camassa-Holm equation are determined by using direct ansatze. As a result, compactons solutions: solitons with the absence of infinite tails, solitons: nonlinear localized waves of infinite support, solitary patterns solutions having infinite slopes or cusps, and plane periodic solutions are established. It is also found that the qualitative change in the physical structure of solutions depends mainly on whether the exponent of the wave function u(x,t) and the expression (2k−c) are positive or negative.