Light beams of the (3+1)D complex Ginzburg–Landau equation induced by the interaction between the external potential and higher-order nonlinearities

In this paper we investigate the impact of the self-steepening (SS) and external potential on the dynamics of the soliton solution of the (3+1)-dimensional cubic–quintic complex Ginzburg–Landau equation, beyond the slowly varying envelope approximation (SVEA) in metamaterials (MMs). In order to create dynamical models with finite degrees of freedom for the description of stable solutions based on the variational approach technique, the Gaussian input with respect to two transverse coordinates (x,y) and the longitudinal coordinate t has been used. The bifurcation diagrams based on the Poincaré map of the periodic motion have been used to analyze the dynamical model. We have obtained a certain number of dynamics, ranging from the stationary, to periodic and chaotic dynamics. With a good choice of dissipative parameters and by increasing the external potential coefficient, it follows that after the transition phase, the soliton transforms into a vortex soliton with a stable dynamics.