Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)8812Research ArticleThe Derivation of the Dispersion Equations of Adiabatic Waveguide Modes in the Thin-Film Waveguide Luneburg Lens in the Form of Non-Linear Partial Differential Equation of the First OrderZuevM ILaboratory of Information Technologieszuev.max@gmail.comAyryanE ALaboratory of Information Technologiesayrjan@jinr.ruBuˇsaJjan.busa@tuke.skIvanovV VLaboratory of Information Technologiesivanov@jinr.ruSevastianovL ATelecommunication System Departmentleonid.sevast@gmail.comStreltsovaO ILaboratory of Information Technologiesstrel@jinr.ruJoint Institute for Nuclear ResearchTechnical University in Ko.sicePeoples’ Friendship University of Russia15042013412213108092016Copyright © 2013,2013This paper presents a derivation of the dispersion equation for a three-layer integrated-optical 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.irregular integrated optical wave guidemethod of adiabatic modescomputer algebra systemнерегулярный интегрально-оптический волноводметод адиабатических модсистемы компьютерной алгебры