A NONLINEAR TRANSMISSION EIGENVALUE PROBLEM THAT DESCRIBES ELECTROMAGNETIC TE WAVE
PROPAGATION IN A PLANE INHOMOGENEOUS NONLINEAR DIELECTRIC WAVEGUIDE 

Valovik Dmitriy Viktorovich, Candidate of physical and mathematical sciences, associate professor, sub-department of mathematics and supercomputer modeling, Penza State University (Penza, 40 Krasnaya str.), dvalovik@mail.ru
Marennikova Ekaterina Alekseevna, Postgraduate student, Penza State University (Penza, 40 Krasnaya str.), mmm@pnzgu.ru
Smirnov Yuriy Gennad'evich, Doctor of physical and mathematical sciences, professor, head of sub-department of mathematics and supercomputer modeling, Penza State University (Penza, 40 Krasnaya str.), mmm@pnzgu.ru 

517.927, 517.968, 519.6 

Objective of the work is to study the mathematical model of surface electromagnetic TE wave propagation in a plane inhomogeneous dielectric waveguide filled with Kerr medium. Material and methods: the physical problem is reduced to a nonlinear integral equation with Green’s function as the kernel. The existence of solutions to the integral equation is proved with the help of the contracting mapping method. For numerical solutions two approaches are suggested: an iteration method (its convergence is proved); the method of Cauchy problem (a variant of the shooting method). Results: the existence of dispersion equation’s roots (propagation constants
of the waveguide) is proved. The authors obtain conditions suitable for k waves propagation. The regions of localization of the propagation constants are found. Conclusions: the results show that there is a nonlinear waveguiding regime for TE waves propagating in a plane inhomogeneous nonlinear waveguide. 

Maxwell’s equations, inhomogeneous waveguide, boundary eigenvalue problem, nonlinear permittivity. 

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