Qualitative analysis and optical soliton solutions galore: scrutinizing the (2+1)-dimensional complex modified Korteweg-de Vries system of equations

Abstract This investigation discusses the (2+1)-dimensional complex modified Korteweg-de Vries (cmKdV) system of equations. The cmKdV equation describes the nontrivial dynamics of water particles from the surface to the bottom of a water layer, providing a more comprehensive understanding of wave behavior. A new version of the generalized exponential rational function method (nGERFM) is utilized to discover diverse soliton solutions. This method uncovers analytical solutions, including exponential function, singular periodic wave, combo trigonometric, shock wave, singular soliton, and hyperbolic solutions in mixed form. Moreover, the planar dynamical system of the concerned equation is created, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore , after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. Numerical simulations of the solutions are added to the analytical results to understand the dynamic behavior of these solutions better. These obtained outcomes provide a foundation for further investigation, making the solutions useful, manageable, and trustworthy for the future development of intricate nonlinear issues. This study’s methodology is reliable, strong, effective, and applicable to various nonlinear partial differential equations (NLPDEs). The Maple software application is used to verify the correctness of all obtained solutions.