Qualitative analysis and new variety of solitons profiles for the (1+1)-dimensional modified equal width equation

The objective of this manuscript is to examine the nonlinear characteristics of the modified equal width equation that is used to simulate the one-dimensional wave propagation nonlinear media, incorporating the dispersion process. Utilizing the traveling wave transformations, we are able to convert the nonlinear partial differential equations (NLPDES) into ordinary differential equations (NLODEs). In this study, an analytical technique is used to utilize the exact soliton solutions of this proposed model. This efficient method is known as the modified auxiliary equation method. This extraction of soliton solutions contains various types of solutions such as trigonometric, hyperbolic, and rational solutions. For a graphical representation, we utilize Mathematica and Maple software to depict the solutions in 3D, 2D, contour plots, and density plots. The main novelty of this paper is to explore the qualitative study, which includes the chaotic behavior, bifurcation, sensitivity, and stability analysis of this problem. For this, first, we apply the Galilean transformation, we convert the NLODEs into two systems of equations. Moreover, the qualitative dynamics of the time-varying dynamical system are examined by employing chaos theory. We explore the intricacies of 3D and 2D phase portraits, time series, and Poincaré maps as powerful tools for detecting the elusive nature of chaos in self-governing dynamic systems. Sensitivity and stability analysis is also studied by using the various initial conditions, revealing the remarkable stability of the system under investigation. The system’s stability is confirmed by the fact that even small changes to the initial conditions have no appreciable effect on the solutions. The results of this study are novel and valuable for further investigation of equations which are helpful for the incoming researchers.