Recent advances in the theory of coupled nonlinear guided waves

The report is devoted to the brief description of the newest results in the theory of coupled nonlinear electromagnetic wave propagation. Throughout the report we consider polarized monochromatic waves and their two-frequency sums in homogeneous medium with nonlinear permittivity (here we discuss only the Kerr law but many results can be extended to the cases of more complicated nonlinearities). Few words are said about linear problems from which the nonlinear ones arise. To be more precise, we begin with guided TE and TM waves propagating in a plane dielectric layer (PDL) with constant permittivity. For want of space in this report we have no opportunity to demonstrate similar results for waveguiding problems in a circular cylindrical dielectric waveguide, however, in the Introduction we give important references. The paper covers very fascinating cases of nonlinear coupled TE-TM and TE-TE wave propagation in a PDL. These coupled waves are two-frequency nonlinear guided waves propagating with two different propagation constants. Main results given here have been obtained by the authors.