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

Abstract This investigation discusses the (2+1)-dimensional complex modified Korteweg–de Vries (cmKdV) system. The cmKdV system 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. The cmKdV system finds applications in various fields of physics and engineering, including fluid dynamics, nonlinear optics, plasma physics, and condensed matter physics. Understanding the behavior predicted by the cmKdV system can lead to insights into the underlying physical processes in these systems and potentially inform the design of novel technologies. 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. To gain a deeper understanding of the dynamic behavior of the solutions, analytical results are supplemented with numerical simulations. 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). As far as we know, this type of research has never been conducted to such an extent for this equation before. The Maple software application is used to verify the correctness of all obtained solutions.