Abstract
The fractional (2+1)-dimensional generalized Boussinesq equation, which incorporates the classical (1+1)- and (2+1)-dimensional Boussinesq and Benjamin-Ono equations as special cases, is investigated symbolically to find the infinitesimal generators and multi-reduction via invariance. The reduced equations are solved using the power series and analytical methods, and at the same time, we acquire a family of the soliton solutions, including singular, combined-singular, breather solitons, bright solitons, and two families of the analytic solutions via two different schemes. With respect to the wave field for the wave dynamics in atmospheric sciences and oceanography, our breather, kink, singular, and bright soliton solutions are closely linked to the nonlinear wave dynamics in the Boussinesq equation, including oscillating wave packets, step-like transitions, and stable localized pulses balanced by nonlinearity and dispersion. The analysis of the fractional parameter $ \beta $'s influence on soliton dynamics through 3D and 2D plots for particular values of parameters shows how $ \beta $ affects the dispersion, amplitude, and intensity of waves. Furthermore, the qualitative behavior of the previously unstudied integer-order equation is investigated through chaotic analysis using a planar dynamical system approach, which has not been explored previously. Key novelties of this work are the derivation of new symmetries, the construction of diverse analytic solutions, and the analysis of the system's dynamical behavior.