In this paper, the variable-coefficient (4+1)-dimensional Fokas (4D-vc Fokas) equation, which describes the evolution of water waves with surface tension in ocean dynamics, is studied using bifurcation analysis. The dynamic system and phase portrait are presented and discussed graphically. Then, the whole discrimination system of the 4D-vc Fokas equation was discussed to classify the possible analytic traveling wave solutions; as a result, novel solitary waves and periodic waves were yielded by setting specific relationships between the coefficients. The soliton wave's motion was affected by the choices of the variable coefficients and took a parabolic and periodic shape; it also became a bright or dark soliton.

In this study, the (4+1)-dimensional Fokas equation with variable coefficients, which describes water waves in deep and wider channels, was reduced to a sixth-order nonlinear ordinary differential equation using the direct similarity reduction method. Then, the Jacobi expansion method was used to obtain multiple novel types of traveling wave solutions, including solitons, periodic waves, and singular waves. The obtained Jacobi wave solutions were considered new and have never been obtained before. Last, the dynamic behavior of the periodic and soliton wave solutions was explained according to different variable coefficient values and visualized by 3D plots.