One approach to the solution of the wave equation for dielectric nonmagnetic media

For dielectric nonmagnetic media, we consider a wave equation obtained from the system of Maxwell's equations in the framework of nonlinear optics with the classical approach. The equation describes the dynamics of the radiation field in glasses, liquids, gases, and many crystals. The dynamics of the electric field of radiation can be analyzed only with the knowledge of the polarization response of the medium to the force action of this field. Therefore, the wave equation can be considered underdetermined. It contains terms depending on the intensity $E=E(x,y,z,t)$ of the electric field of the radiation and a term depending on the polarization response of the medium $P=P(x,y,z,t)$. We propose a method for solving this underdetermined equation. Since the polarization response of the medium is caused by the force action of the electric field of radiation, we assume that the equation $P=P(E)$, $E(x,y,z,t)=\rm{const}$, defines a level surface of the function $P$. Under this assumption, the wave equation reduces to an ODE. The independent variable in the ODE is the function $E$. The function $E=E(x,y,z,t)$ is found by solving the first-order partial differential equation (the basic equation) $E_{t}=f_{0}(E)$. The solution of the ODE and the form of $E=E(x,y,z,t)$ (hence, the dynamics of the radiation field) depend on the choice of an arbitrary function $f_{0}(E)$. The form of $E=E(x,y,z,t)$ and $P=P(E)$ is written for four functions $f_{0}(E)$. These solutions are found up to an arbitrary constant.