Bifurcation analysis and novel traveling wave solutions for the (4+1)-dimensional Fokas equation with variable coefficients

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.