The Analysis of Bifurcation, Quasi-Periodic and Solitons Patterns to the New Form of the Generalized q-Deformed Sinh-Gordon Equation

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.