Considering Traveling Waves and the Dynamical Behavior of the Complex Coupled Kairat-Zhaidary-Myrzakulov System

This article presented a formal systematic transformation of the complex coupled Kairat-Zhaidary-Myrzakulov system (KZMS) into a single ordinary differential equation (ODE), with the aim of providing a complete dynamical picture of the system that goes beyond the mere construction of exact solutions. The examined model is an essential integrable system that has applications in nonlinear optics and plasma physics. We selected the fitting traveling wave transformation structure to perform this process as well. The reduction process provided explicit links between the field parameters and variables. A single second-order ODE is the output of this transformation. Results were obtained by using two analytically methods, specifically the extended rational sin-cos (ERSC) method and the modified version of the generalized Kudryashov method (MGKM). Additionally, phase portraits and dynamic behaviors of the obtained solutions were discussed. To best understand the physical characteristics of the outcomes, we plotted them in different kinds of figures with three and two dimensions. To confirm the accuracy of the work, all obtained solutions were substituted into the original model, and their validity was verified. Computational tools were used for this assignment. Finally, a detailed phase portrait and bifurcation analysis were performed to understand the stability properties and dynamical changes of the system.