Criterion of existence of longitudinally inhomogeneous traveling waves in nonlinear electrodynamics

The criterion referred to in the title is established solely on the basis of the behavior of the function $\varepsilon(u)u^4$, where $u$ is the amplitude of the field in question and $\varepsilon(u)$ is the nonlinear permittivity of the medium involved. The criterion is used to discuss the possibility of realization of one-dimensional longitudinally inhomogeneous waves under various conditions and, in particular, in nonlinear optics.