A simplified mathematical model is proposed to describe the dynamics of a quasi-monochromatic light wave in the bulk of an effectively isotropic metamaterial with a nearly zero averaged permittivity in the presence of weak spatial inhomogeneity, Kerr nonlinearity, and linear gain under external pumping. The model is based on a general Ginzburg–Landau vector equation with the dominance of the curl–curl term in the dispersion operator and resembles the equation for electromagnetic waves in a plasma E.A. Kuznetsov, “The Collapse of Electromagnetic Waves in a Plasma,” Sov. Phys. JETP 39 , 1003 (1975). The split step Fourier method is appropriated in the case of purely real Kerr coefficients. This method has made it possible to reveal various variants of non-trivial evolution of both centrosymmetric and toroidal vector wave structures in a parabolic well potential, as well as the nonlinear interaction between longitudinal and transverse waves in the case of a combination of these structures.
V. P. Ruban (2026) studied this question.