Fundamental and regular transport solutions of Maxwell's equations and their properties at superluminal speeds: shock electromagnetic waves

Transport solutions of Maxwell's equations under the action of moving emitters of electromagnetic waves moving with a constant velocity in a fixed direction are considered. Using the Fourier transform of generalized functions, fundamental and generalized solutions are constructed for velocities exceeding the velocity of electromagnetic wave propagation in a medium and coinciding with it, which is called the light velocity. Their regular integral representations are given in analytical form. Construction of solutions for arbitrary moving sources is based on the property of convolution of fundamental solutions of differential equations with the right-hand side. It is shown that shock electromagnetic waves arise at such velocities. Using the method of generalized functions, conditions are obtained for jumps in electromagnetic field strengths at shock wave fronts. It is shown that shock electromagnetic waves are transverse, and the vectors of electric and magnetic strength are orthogonal to each other and lie in the tangent bundle to the shock wave front.