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This paper is subjected to study the nonlinear integrable model which is the (3+1)-dimensional Boussinesq equation which has a lot of applications in engineering and modern sciences. To find and examine the analytical exact solitary wave solutions of (3+1)-dimensional Boussinesq equation, a modified generalized exponential rational functional method is exerted. As a result, waves, singular periodic, hyperbolic, and trigonometric type solutions are obtained. These acquired solutions are more innovative and encouraging to researchers in their endeavor to study physical marvels. To illustrate how some selected exact solutions propagate, the graphical representation in 2D, Contour, and 3D of those solutions is provided with various parametric values. The considered equation is additionally transformed into the planar dynamical structure by applying the Galilean transformation. All potential phase portraits of the dynamical system are investigated using the theory of bifurcation. The Hamiltonian function of the dynamical system of differential equations is established to see that, the system is conservative over time. The presentation of energy levels through graphics provides valuable insights, and it demonstrates that the model has solutions that can be expressed in closed form. The periodic, quasi-periodic, and chaotic behaviors of the 2D, 3D, and time series are also observable once the dynamical system is subjected to an external force. Meanwhile, the sensitivity of the derived solutions is carefully examined for a range of initial conditions. [ABSTRACT FROM AUTHOR]

The renowned non-linear Kairat-II equation, which expresses the surface curves, is examined meticulously in this work. There has never been a study before that discussed conserved quantities, sensitivity analysis, Hamiltonian function, Lie symmetry invariance criteria, and invariant solutions of Kairat-II equation The Lie invariance criteria are taken into consideration by the symmetry generators. The suggested technique leads to in a four-dimensional Lie algebra, where the translation point symmetries in space and time correlate with the conservation of mass and the energy, respectively, and remaining point symmetries are dilation and scaling. Firstly, we use symmetry reduction of Lie subalgebras to obtain closed-form invariant solutions. In specific reduction cases, we transform the Kairat-II equation into a spectrum of non-linear ordinary differential equations, which have the benefit of being able to offer an extensive selection of closed-form solitary wave solutions. For this equation, the Cauchy problem cannot be solved by the inverse scattering transform, therefore, traveling wave exact solutions are generated using the analytical Paul-Painlevé approach. In both two and three dimensions, the graphical behavior of specific solutions is presented for specific quantities of physical factors of examined equation. Utilizing conservation laws multipliers, the study culminates by determining an extensive set of local conservation laws for the non-linear Kairat-II equation that are applicable to arbitrary constant coefficients. Instead of focusing on the physical aspects of conservation laws, this study computed the conserved quantities using a mathematical perspective that can be used to identify potential symmetries of partial differential equations, guiding readers toward the solutions of partial differential equations. The existence of the Hamiltonian function is presented. The model's sensitivity to various starting conditions is highlighted by the sensitivity analysis. • Construction of Lie point symmetries. • Formation of invariant exact analytical solutions. • The development of Hamiltonian function for dynamical system. • The sensitivity analysis performed. • The first conserved quantities developed. [ABSTRACT FROM AUTHOR]