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ISRN Mathematical Physics
Volume 2013 (2013), Article ID 184325, 7 pages
Research Article

On the Problem of Electromagnetic Waves Propagating along a Nonlinear Inhomogeneous Cylindrical Waveguide

Department of Mathematics and Supercomputing, Penza State University, Krasnaya Street. 40, Penza 440026, Russia

Received 23 April 2013; Accepted 6 June 2013

Academic Editors: G. Cleaver, J. Garecki, F. Sugino, and G. F. Torres del Castillo

Copyright © 2013 Yury G. Smirnov and Dmitry V. Valovik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Electromagnetic TE wave propagation in an inhomogeneous nonlinear cylindrical waveguide is considered. The permittivity inside the waveguide is described by the Kerr law. Inhomogeneity of the waveguide is modeled by a nonconstant term in the Kerr law. Physical problem is reduced to a nonlinear eigenvalue problem for ordinary differential equations. Existence of propagating waves is proved with the help of fixed point theorem and contracting mapping method. For numerical solution, an iteration method is suggested and its convergence is proved. Existence of eigenvalues of the problem (propagation constants) is proved and their localization is found. Conditions of k waves existence are found.