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.