Similarities in a fifth-order evolution equation with and with no singular kernel

We perform in this report a comparative analysis between differential fractional operators applied to the non-linear Kaup–Kupershmidt equation. Such operators include the Atangana–Beleanu derivative and the Caputo–Fabrizio derivative which respectively follow the Mittag-Leffler law and the exponential law. We exploit the fixed points of the dynamics and the stability analysis to demonstrate that the exact solution exists and is unique for both types of models. Methods of performing numerical approximations of the solutions are presented and illustrated by graphical representations exhibiting a clear comparison between the dynamics under the influence of Mittag-Leffler law and those under the exponential law. Different cases are presented with respect to values of the derivative order 0 < α ≤ 1. We note a slight difference between both dynamics in terms of individual points, but their global pictures remain similar and close to the traditional and popular traveling wave solution of the standard Kaup–Kupershmidt model (α=1).