On the fractal dynamics for higher order traveling waves

Abstract

Auto replication processes remain fascinating in sciences, engineering and technology as their applications in machining/biological systems have been widely used to solve number of outstanding problems in communities’ every day lives. Finding innovative techniques capable of generating auto replication processes in various fields has then become the priority for number of scientists. One of those fields includes wave motion. In this paper, we use the 7th order Korteweg–de Vries (KdV) model, combined with the fractal-fractional operator to artificially (numerically) generate auto replication processes characterizing the evolution of higher order traveling waves. The well-posedness for the combined model is first studied with the establishment of its existence and uniqueness results. Numerical simulations then follow and prove that the higher order traveling wave can be involved in a self replication process. There is generation of the exact or approximately exact copies of the initial traveling wave in different scales and where the fractal process produces other multiple traveling waves that look like the preceding ones. Furthermore, the fractal dynamics expand as the model’s parameter γ changes. We are on the right track for the artificially-formed fractal process of higher order traveling waves applicable in wave motion’s domain.