Bifurcation analysis and traveling wave solutions of time fractional Zakharov-Kuznetsov equation

Differential equations have an important place in applied sciences. They have been widely used in solving various problems, especially complex problems. The ZK equation is a partial differential equation used especially in plasma phiysics and in the study of waves in magnetized plasma. In this study, the exact solutions of the nonlinear (2+1) dimensional ZK equation given with the Riemann-Liouville (R-L) time fractional derivative operator were obtained by Generalized Exponential Rational Function Method (GERFM) and the Sub-Equation Method. The given partial differential equation was transformed into an ordinary differential equation with the appropriate wave transformation. By finding the homogeneous equilibrium constant, exact solutions were sought in accordance with the homogeneous equilibrium principle, and the necessary parameters were found by solving with symbolic calculation programs. In dynamical systems theory, the phenomenon of bifurcation has been extensively studied in the literature from many perspectives. Bifurcation is a phenomenon in which a slight parametric perturbation in a dynamical system produces qualitative changes in the structure of the solutions. It can be interpreted as a bifurcation when, due to a slight parameter change, the stable equilibrium of the differential equations suddenly becomes unstable and a stable periodic orbit appears near the equilibrium. To analyze such phenomena, bifurcation theory has been studied and applied to systems biology, synthetic biology, power system, epidemic disease model analysis, etc. It is widely used for the analysis and synthesis of complex behavior in many research areas such as. In this study, the dynamic system phase plane analysis of the (2+1) dimensional equation was studied. . First, by fractional transformation, the partial differential equation was reduced to a second-order ordinary differential equation and written as a dynamic system. Critical points were determined by establishing the Jakobi determinant. Phase plane analysis of the system was carried out by considering different situations of the parameters in the equation. Again, phase plane graphs were drawn for different states of the parameters.