BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS FOR A GENERALIZED KLEIN- GORDON EQUATION WITH BETA FRACTIONAL DERIVATIVES

Abstract

In this paper, we consider the exact traveling wave solutions of the generalized Klein-Gordon equation with beta fractional derivatives. By using the bifurcation theory of dynamical systems, we derive the bifurcation behavior of its phase portraits under various parameter settings. For some special level curves, we obtain all possible traveling wave solutions such as periodic wave solutions, dark and bright solitary wave solutions, kink and anti-kink solutions, and some unbounded wave solutions. The impacts of beta fractional orders are analyzed using graphs, with a focus on their impact on temporal delay, broadening, and the oscillation period. Furthermore, the present work generalizes and extends earlier findings in the literature. These results contribute to a deeper understanding of the dynamics of the generalized Klein-Gordon equation model with beta fractional derivatives.