Fractional Kuramoto–Sivashinsky equation with power law and stretched Mittag-Leffler kernel

The presence of a complex spatio-temporal behavior in spatially extended systems (SES) as a result of several mechanisms that interact non-linearly with other nearby has attracted a lot of attention in recent decades. A well-known example of SES is the Kuramoto–Sivashinsky (KS) equation. In the search for a broader perspective of some unusual irregularities observed in the context of phase turbulence in the reaction–diffusion systems, the propagation of the wrinkled flame front and the unstable drift waves driven by the collision of electrons in a Tokamak, we explored the possibility of extending the analysis of the KS equation with three perturbation levels using the conceptions of fractional differentiation with non-local and non-singular kernel. We use the fractional order operator of Atangana–Baleanu in the sense of Liouville–Caputo (ABC) to set the fractional model of KS that we analyze in this article. We prove existence and uniqueness of a continuous solution to fractional KS model, using the fixed-point theorem and the Picard–Lindelöf approach, provided conditions on the perturbation parameters and the order of the fractional operator. In addion, using the theory of fixed point, we present the stability of the iterative method generated by the Picard–Lindelöf approach. Finally, we presented the approximate analytical solution of the fractional KS equation using the homotopy perturbation transform method. Some numerical simulations are carried out for illustrating the results obtained.