On a class of vector fields

It is shown that a simple postulate ``The displacement field of the vacuum is a normalized electric field'', is equivalent to three parametric representation of the displacement field of the vacuum: $$ u(x;t) = P(x) \cos k(x)t + Q(x) \sin k(x)t. $$ Here $t$ is time; $k(x)$ -- frequency vibrations at the point of three-dimensional Euclidean space; $P(x), Q(x)$ -- a pair of stationary orthonormal vector fields; $(k,P, Q)$ -- parameter list of the displacement field. In this case, the normalization factor has dimension $T^{-2}$. The speed of the displacement field $$ v(x;t) = \frac{\partial u(x;t)}{\partial t} = k(x)(Q(x) \cos k(x)t - P(x) \sin k(x)t). $$ The electric field corresponding to this distribution of the displacement field of vacuum, is given by the formula $$ E(x;t) = -\frac{\partial v(x;t)}{\partial t} = k^2(x)u(x;t). $$ Moreover, the magnetic induction $$ B(x;t) = \mathop{\mathrm{rot }} v(x; t). $$ These constructions are used in the determination of local and global solutions of Maxwell's equations describing the dynamics of electromagnetic fields.

local and global solutions of Maxwell’s equations, spectral problem for rotor operator, the small flow of the displacement field