In this manuscript, a new form of the generalized q-deformed Sinh-Gordon equation is investigated which could model physical systems with broken symmetries and to incorporate phenomena involving amplification or dissipation. The proposed model is explored based on the Lie symmetry approach. Using similarity reduction, the partial differential equation is transformed into an ordinary differential equation. By employing the generalized auxiliary equation approach, precise results for the derived equation are obtained. The solutions are graphically depicted as 3D, 2D, and contour plots. Furthermore, the qualitative analysis of the considered model is investigated by employing the concepts of bifurcation and chaos. The phase profiles are displayed for different sets of the parameters. Additionally, by applying an external periodic strength, quasi-periodic and chaotic behaviors are documented. Various tools for detecting chaos are discussed, including 3D and 2D phase patterns, time series, and Poincaré maps. Additionally, a sensitivity analysis is conducted for various initial conditions. The obtained findings are unique and indicate the viability and efficacy of the suggested strategies for evaluating soliton solutions and phase illustrations for various nonlinear models.

This study explores the modified Oskolkov equation, which depicts the behavior of the incompressible viscoelastic Kelvin–Voigt fluid. The primary focus of this research lies in several key areas. Firstly, the Lie symmetries of the considered equation are identified. These symmetries are utilized to transform the discussed model into an ordinary differential equation. Analytical solutions are subsequently derived using the new auxiliary equation technique. Next, a comprehensive analysis of the equation’s dynamic nature is undertaken from multiple aspects. Bifurcation is carried out at fixed points within the system, and chaotic behavior is unveiled by introducing an external force to the dynamic system. Various tools, including 3D and 2D phase plots, time series, Poincaré maps, and multistability analysis, are employed to identify the chaotic nature of the system. Furthermore, the sensitivity of the model is explored across diverse initial conditions. In general, comprehending the dynamic characteristics of systems holds immense significance in forecasting outcomes and innovating new technologies.

In this manuscript, the Landau-Ginzburg-Higgs (LGH) equation is considered as an investigating model. To extract novel results from the governing equation, the $ G'/(b G'+G+a) $-expansion approach has been employed. Utilizing this approach, the outcomes are attained as hyperbolic and trigonometric functions. Kink, periodic and singular soliton solutions have been recovered by selecting the appropriate values for the parameters. The obtained findings for the LGH equation are displayed in 3-D, contour and 2-D profiles. Using Galilean transformation, the model is converted into a planar dynamical system, and qualitative analysis is investigated. Moreover, chaotic and quasi-periodic patterns have been addressed after including the perturbed term. Simulated results reveal that by modifying amplitude and frequency parameters, the dynamic behavior of the system can also be changed. The recorded results are novel and show the effectiveness and feasibility of the suggested technique for assessing soliton solutions and phase visualizations for different nonlinear models.