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An extension of the nonlinear Schrodinger equations was introduced by Laskin in , in the form of a fractional equation:As is known, solitons are eigenmodes of problems arising in dispersive nonlinear media. Examples of such problems can be Bose-Einstein condensates, optical waveguides and a system of carbon nanotubes. Note that the latter system was considered in the approximation when carbon nanotubes are evenly distributed over the sample volume. In addition, an equation was obtained for carbon nanotube systems, which is a generalization of the nonlinear Schrodinger equation. Nonlinear Schrodinger equations (NUS), with focusing or defocusing nonlinearity, are universal models that make it possible to consider solitons of all personal types. Varieties of solitons can be radically supplemented in spatially inhomogeneous conditions . Usually nonlinear lattices are used, i.e. periodically spatially modulated changes in the nonlinearity coefficient.

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Currently, much attention is paid to the study of photonic crystals – materials with an ordered structure characterized by a strictly periodic change in the refractive index at scales comparable to the wavelengths of radiation in visible and near infrared ranges. This is a dynamically developing direction of modern materials science. It is connected with the possibility of creating LEDs with high efficiency, new types of lasers with low threshold generation, light waveguides, optical switches, filters, as well as digital computing devices based on Photonics. The aim of this work is to calculate the reflection and transmission of a polarized light wave from a layered system of nanostructures that form a periodic medium. The calculation is carried out by two methods: the method of characteristic matrices and the method based on the use of Chebyshev polynomials. The authors have created a basic component of the computer program for calculating the reflection coefficient and the transmittance of layered nanostructures. The paper calculates the spectra of reflection and transmission coefficients and presents the analysis of the results obtained. The basic element is chosen as a basic nanostructure: a layer of magnesium oxide MgO 100 nm thick, a diamond layer 160 nm thick, an arsenic layer AsBr3 tribromide 80 nm thick, a silicon layer 120 nm thick. The authors compare the two methods used: the results are almost the same, which makes it possible in practice for such structures to use a simpler method for the computational procedure based on Chebyshev polynomials.