Geodesic
Nonlinear transmission eigenvalue problem describing TE wave propagation in two-layered cylindrical dielectric waveguides
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 7, pp. 1150-1161

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The paper is concerned with propagation of surface TE waves in a circular nonhomogeneous two-layered dielectric waveguide filled with a Kerr nonlinear medium. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of a Green’s function. The existence of propagating TE waves is proved using the contraction mapping method. For the numerical solution of the problem, two methods are proposed: an iterative algorithm (whose convergence is proved) and a method based on solving an auxiliary Cauchy problem (the shooting method). The existence of roots of the dispersion equation (propagation constants of the waveguide) is proved. Conditions under which k waves can propagate are obtained, and regions of localization of the corresponding propagation constants are found. 
@article{ZVMMF_2013_53_7_a10,
     author = {D. V. Valovik and Yu. G. Smirnov and E. Yu. Smol'kin},
     title = {Nonlinear transmission eigenvalue problem describing {TE} wave propagation in two-layered cylindrical dielectric waveguides},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1150--1161},
     publisher = {mathdoc},
     volume = {53},
     number = {7},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_7_a10/}
}
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D. V. Valovik; Yu. G. Smirnov; E. Yu. Smol'kin. Nonlinear transmission eigenvalue problem describing TE wave propagation in two-layered cylindrical dielectric waveguides. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 7, pp. 1150-1161. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_7_a10/