The evolution of nonmonochromatic mode fields in a waveguide with space dispersive medium

We consider a cylindrical waveguide with layered space dispersive medium: there are no free source and charges. The space dispersion is taken in the constitutive equations as a dependence of the electric induction on the electric field and its space derivatives. Electromagnetic fields are; represented as series of nonmonochromatic mode E- and H- fields. In studying the solvability of evolutionary waveguide equations, we obtain a partial differential equation of Sobolev type. We investigate this equation and develop a new numerical method