Phase characterization and exraction of new forms of solitons for the (3+1)-dimensional q-deformed Sinh-Gordon equation

In this article, the (3+1)-dimensional q-deformed Sinh-Gordon model is investigated to extract analytical solutions using the unified method. This technique effectively extracts polynomial and rational function solutions. When the appropriate limiting constraints are given to the parameters, this technique successfully retrieves hyperbolic and trigonometric results. Some graphical representations of the solutions of the proposed equation are illustrated. Additionally, all feasible phase portraits are shown, the planer dynamical system of the equation under discussion is built using Galilean transformation, and sensitive inspection is used to verify the sensitivity of the equation under consideration. There aren't many previous methods for solving this kind of equation, either analytically or numerically. This work is highly valuable for the understanding of various symmetrical physical systems.