Around Chaotic Disturbance and Irregularity for Higher Order Traveling Waves

Emile F Doungmo Goufo,Ignace Tchangou Toudjeu

doi:10.1155/2018/2391697

Emile F Doungmo Goufo, Ignace Tchangou Toudjeu

Open Access

https://doi.org/10.1155/2018/2391697

Copy DOI

Abstract

Many unknown features in the theory of wave motion are still captivating the global scientific community. In this paper, we consider a model of seventh order Korteweg–de Vries (KdV) equation with one perturbation level, expressed with the recently introduced derivative with nonsingular kernel, Caputo-Fabrizio derivative (CFFD). Existence and uniqueness of the solution to the model are established and proven to be continuous. The model is solved numerically, to exhibit the shape of related solitary waves and perform some graphical simulations. As expected, the solitary wave solution to the model without higher order perturbation term is shown via its related homoclinic orbit to lie on a curved surface. Unlike models with conventional derivative (γ=1) where regular behaviors are noticed, the wave motions of models with the nonsingular kernel derivative are characterized by irregular behaviors in the pure factional cases (γ<1). Hence, the regularity of a soliton can be perturbed by this nonsingular kernel derivative, which, combined with the perturbation parameter ζ of the seventh order KdV equation, simply causes more accentuated irregularities (close to chaos) due to small irregular deviations.

Highlights

In the last decade, a great number of researchers have paid a particular attention to the study of solitary wave equations that undergo the influence of external perturbations
Korteweg–de Vries (KdV) equation and its variants are of infinite dimension and their use to address traveling waves or chaotic dynamics of low dimension is facilitated by numerical approximations, which have proven that correlation dimension established via Grassberger-Procaccia technique and information dimension obtained from formula of Kaplan-Yorke are both between two and three for steady traveling waves [3]
As a response to the Caputo-Fabrizio fractional order derivative (CFFD) and being aware of the conflicting situations that exist between the classical Riemann-Liouville and Caputo derivatives, the classical Riemann-Liouville definition was modified [9, 10] in order to propose another definition known as the new RiemannLiouville fractional derivative (NRLFD) without singular kernel and expressed for γ ∈ [0, 1] as aDγt u (t)

Summary

Introduction

A great number of researchers have paid a particular attention to the study of solitary wave equations that undergo the influence of external perturbations. In this paper, the same trend of numerical approach by making use of the recently developed fractional derivative with nonsingular kernel [8,9,10,11,12,13], to express a seventh order Korteweg–de Vries (KdV) equation with one perturbation level. This is the first instance where such a model is extended to the scope of fractional differentiation and fully investigated. We shall give a brief review of the recent developments done in the theory of fractional differentiation

Around the Nonsingular Kernel Differentiation with Fractional Order

Existence and Uniqueness

Shape of Solitary Waves via Numerical Approximations

Concluding Remarks