Zero refractive index metamaterial is a kind of artificial structure material with zero refractive index, which has many unique applications in controlling the propagation of light and other electromagnetic waves [1]. Nonlinear materials can adaptively regulate the ability of electromagnetic waves under specific conditions, combining zero-refractive-index metamaterials with nonlinear effects, we can get this metamaterial with many unique and practical functions, such as superlens, wavelength conversion and frequency modulation, smart materials and optical switches. The behavior of electromagnetic wave propagation in nonlinear zero-refractive-index metamaterials can be controlled by a nonlinear system of Maxwell’s equations. \( \varepsilon \varvec{E}_{t}+\sigma _{1} (x,|\varvec{E}|)\varvec{E}=\nabla \times \varvec{H},\quad \mu \varvec{H}_{t} +\sigma _{2} (x,|\varvec{H}|)\varvec{H}=-\nabla \times \varvec{E} + f( |\varvec{H}|)\varvec{H} \) where \(\sigma _{1} (x,s)s,\sigma _{2} (x,s)s\) is a monotonic function on s, and \(f(|\varvec{H}|)\varvec{H}\) denotes the magnetic current that varies with the strength of \(\varvec{H}\) . In this paper, we establish and estimate the convergence of the consistent stable solution of the nonlinear Maxwell system to the solution of the proposed smooth system and its corresponding rate of convergence when the dielectric function and the magnetic permeability function converge to zero simultaneously. The lack of tightness in the equations makes the lack of tight embedding in the proof of the existence of weak solutions, which in turn makes the nonlinear terms more difficult to handle. We overcome this difficulty by obtaining an estimate of the \(H^{1}\) -paradigm of \(\varvec{H}\) . Other key methods include energy estimation.
