Exact solutions to the (1+1)-dimensional nonlinear Maxwell equations in the orthogonal curvilinear coordinates

Characterizing electromagnetic wave propagation in nonlinear and inhomogeneous media is of great interest from both theoretical and practical perspectives, even though it is extremely complicated. In fact, it is still an unresolved issue to find the exact solutions to the nonlinear waves in the orthogonal curvilinear coordinates. In this paper, we present an analytic method to handle the problem of electromagnetic waves propagation in arbitrarily nonlinear and particularly inhomogeneous media without dispersion. Through the exact solutions of the (1+1)-dimensional nonlinear Maxwell equations, we discuss some nonlinear phenomena, including cylindrical shock waves, free nonlinear oscillations, and nonlinear superposition of waves.