UDC 517:519.6, 535+537.8:621.37 The Derivation of the Dispersion Equations of Adiabatic M. I. Zuev∗, E. A. Ayryan∗, J. Buˇ sa†, V. V. Ivanov∗, L. A. Sevastianov‡, O. I. Streltsova∗ ∗ Laboratory of Information Technologies Joint Institute for Nuclear Research Joliot-Curie 6, 141980 Dubna, Moscow region, Russia † Technical University in Koˇ sice Letn´ a 9, 04001, Koˇ sice, Slovak Republic ‡ Telecommunication System Department Peoples’ Friendship University of Russia Miklukho-Maklaya str. 6, 117198 Moscow, Russia This paper presents a derivation of the dispersion equation for a three-layer integratedoptical Luneburg lens based on the method of adiabatic waveguide modes. <...> From this equation there follows the relationship between the coefficient of phase deceleration and function, which determines the thickness of the irregular waveguide layer. <...> The dispersion equation is represented in the form of non-linear partial differential equation of the first order with coefficients, depending on parameters. <...> Among these parameters are regular waveguide layer thickness and optical parameters of the pending Luneburg lens. <...> To represent the dispersion equation in the form of differential equations in partial derivatives, it is necessary to calculate a symbolic form the determinant of a matrix of 12th order, which determines the solubility of the system of linear algebraic equations, resulting from the boundary conditions. <...> To calculate this determinant in analytical form a procedure of reduction of the system of linear algebraic equations with the use of the computer algebra system Maple is proposed. <...> Key words and phrases: irregular integrated optical wave guide, method of adiabatic modes, computer algebra system. 1. <...> One of the most promising methods used in the modeling of the waveguide propagation of radiation with the exact tangential boundary conditions, is the method of adiabatic waveguide modes. <...> The overall objective of modeling smoothly irregular longitudinally integrated opThe waveguide Luneburg lens is an important functional element of integrated Waveguide Modes in the Thin-Film Waveguide Luneburg Lens in the Form of Non-Linear Partial Differential Equation of the First Order tical waveguides includes the task of finding the irregular surface of the waveguide layer by solving the dispersion equation. <...> In the papers [3–7] computational scheme for solving this problem has been proposed. <...> However, to develop a software module that allows to increase significantly <...>