Bifurcations and Exact Solutions for a Class of MKdV Equations with the Conformable Fractional Derivative via Dynamical System Method

Abstract

In 2014, Khalil et al. [2014] proposed the conformable fractional derivative, which obeys chain rule and the Leibniz rule. In this paper, motivated by the monograph of Jibin Li [Li, 2013], we study the exact traveling wave solutions for a class of third-order MKdV equations with the conformable fractional derivative. Our approach is based on the bifurcation theory of planar dynamical systems, which is much different from the simplest equation method proposed in [Chen & Jiang, 2018]. By employing the traveling wave transformation [Formula: see text] [Formula: see text], we reduce the PDE to an ODE which depends on the fractional order [Formula: see text], then the analysis depends on the order [Formula: see text]. Moreover, as [Formula: see text], the exact solutions are consistent with the integer PDE. However, in all the existing papers, the reduced ODE is independent of the fractional order [Formula: see text]. It is believed that this method can be applicable to solve the other nonlinear differential equations with the conformable fractional derivative.