Bifurcation Analysis and Implicit Solution of Klein-Gordon Equation with Dual-power Law Nonlinearity in Relativistic Quantum Mechanics

Abstract

This paper focuses on the nonlinear Klein-Gordon equation with dual power law nonlinearity. The traveling wave hypothesis reveals an implicit solution in terms of Gauss' hypergeometric functions. The bifurcation analysis is subsequently carried out that gives an additional set of solutions. The phase portraits of the solutions are also given.

New Exact Solutions, Dynamical and Chaotic Behaviors for the Fourth-Order Nonlinear Generalized Boussinesq Water Wave Equation

“…Then, we obtain det MðP 1 Þ = −β and det MðP 2 Þ = β. By the theory of planar dynamical systems [24][25][26][27][28][29][30], we draw the following conclusion as in Table 1. The different phase portraits of system (50) are shown in Figures 1 and 2.…”

“…The analysis of bifurcation and chaos behavior is a very interesting nonlinear phenomenon, which has been applied in many fields, such as engineering, telecommunication, and ecology [24][25][26][27]. By analyzing the dynamic behavior of differential equation, we can study whether the periodic external perturbation will lead to the chaotic behavior of differential equation.…”

Nonlinear dynamics of a generalized higher-order nonlinear Schrödinger equation with a periodic external perturbation

“…In this part, we investigate the bifurcation and chaotic motions of Eq. (2) which are the interesting nonlinear phenomena and have great applications in many fields such as the technological, engineering, telecommunications, ecology [27,28]. The addition of a perturbation can lead to the system non-integrable.…”

“…When Γ = 0, Eq. (1) can be used to describe the slowly varying electromagnetic waves in optical fibers [14,27,28]. However, for the ultrashort pulses propagation in the high-bit-rate and long-distance communication, some higher-order linear and nonlinear terms have to be incorporated into the NLS equation [14].…”

Extension on peakons and periodic cusp waves for the generalization of the Camassa-Holm equation

“…In our previous work , we studied the peakons and periodic cusp wave solutions of Eq. when n = 1, m ≥2 and n ≥2, m = n + 1, by exploiting the bifurcation method and qualitative theory of dynamical systems , and setting the integral constant to be zero. The results of are summarized as follows: When n = 1, m = 2, Eq.…”