Phase-Plane Analysis of Solutions of the Helmholtz Equation for Electromagnetic Waves in Media with Self-Action

We describe the application of the phase plane method to analyze the behavior of stationary harmonic plane electromagnetic waves. The phase plane method allows one to describe qualitatively all possible solutions of the Helmholtz nonlinear differential equation without performing a numerical integration. All types of phase portraits for waves in the media with self-action are presented and their physical interpretation is discussed. The consideration is carried out on the model of cubic nonlinearity without specifying the physical mechanism of this nonlinearity. The features of the nonlinear case are discussed. The phenomenon of the self-limitation of the transmitted wave power in a medium with negative nonlinearity is described.