Geodesic
On the problem of propagation of nonlinear coupled TE–TM waves in a layer
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 3, pp. 504-518 
D. V. Valovik; Yu. G. Smirnov. On the problem of propagation of nonlinear coupled TE–TM waves in a layer. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 3, pp. 504-518. http://geodesic.mathdoc.fr/item/ZVMMF_2014_54_3_a13/
@article{ZVMMF_2014_54_3_a13,
     author = {D. V. Valovik and Yu. G. Smirnov},
     title = {On the problem of propagation of nonlinear coupled {TE{\textendash}TM} waves in~a~layer},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {504--518},
     year = {2014},
     volume = {54},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2014_54_3_a13/}
}
TY  - JOUR
AU  - D. V. Valovik
AU  - Yu. G. Smirnov
TI  - On the problem of propagation of nonlinear coupled TE–TM waves in a layer
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2014
SP  - 504
EP  - 518
VL  - 54
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2014_54_3_a13/
LA  - ru
ID  - ZVMMF_2014_54_3_a13
ER  - 
%0 Journal Article
%A D. V. Valovik
%A Yu. G. Smirnov
%T On the problem of propagation of nonlinear coupled TE–TM waves in a layer
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2014
%P 504-518
%V 54
%N 3
%U http://geodesic.mathdoc.fr/item/ZVMMF_2014_54_3_a13/
%G ru
%F ZVMMF_2014_54_3_a13

Voir la notice de l'article provenant de la source Math-Net.Ru

The problem of simultaneous propagation of two types of electromagnetic waves (TE and TM) in a plane dielectric waveguide filled with a nonlinear medium is considered. These polarized waves have different frequencies and different propagation constants. The physical problem is reduced to a nonlinear two-parameter transmission eigenvalue problem for Maxwell’s equations in a layer. The coupled eigenvalues are coupled propagation constants. A theorem on the existence and localization of coupled eigenvalues corresponding to coupled polarized electromagnetic waves is proved.

[1] Atkinson F. V., Mingarelli A. B., Multiparameter eigenvalue problems. Sturm-Liouville theory, CRC Press, NW, 2011 | MR | Zbl

[2] Abramov A. A., “Metod resheniya nekotorykh mnogoparametricheskikh zadach na sobstvennye znacheniya, voznikayuschikh pri ispolzovanii metoda Fure”, Zh. vychisl. matem. i matem. fiz., 34:10 (1994), 1524–1525 | MR

[3] Valovik D. V., “Zadacha o rasprostranenii elektromagnitnykh TM-voln v sloe s proizvolnoi nelineinostyu”, Zh. vychisl. matem. i matem. fiz., 51:9 (2011), 1729–1739 | MR | Zbl

[4] Valovik D. V., Smirnov Yu. G., Rasprostranenie elektromagnitnykh voln v nelineinykh sloistykh sredakh, Izd-vo PGU, Penza, 2010

[5] Zhidkov P. E., “O bazisnosti Rissa sistemy sobstvennykh funktsii nelineinoi zadachi tipa Shturma–Liuvillya”, Matem. sb., 191:3 (2000), 43–52 | DOI | MR | Zbl

[6] Boardman A. D., Egan P., Lederer F. et al., “Third-order nonlinear electromagnetic TE and TM guided waves”, Nonlinaer surface electromagnetic phenomena, Modern problems in condensed matter sciences, 29, eds. Ponath H.-E., Stegeman G. I., Elsevier, Norht-Holland, 1991

[7] Smirnov Yu. G., Valovik D. V., “Coupled electromagnetic TE-TM-wave propagation in a layer with Kerr nonlinearity”, J. Math. Phys., 53:12 (2012), 123530, 24 pp. | DOI | Zbl

[8] Eleonskii V. M., Oganesyants L. G., Silin V. P., “Vektornaya struktura elektromagnitnogo polya v samosfokusirovannykh volnovodakh”, Uspekhi fiz. nauk, 107:3 (1972), 516–518 | DOI

[9] Eleonskii V. M., Oganes'yants L. G., Silin V. P., “Structure of three-component vector fields in self-focusing waveguides”, Soviet Phys. JETP, 36:2 (1973), 282–285 | MR

[10] Valovik D. V., “On the problem of nonlinear coupled electromagnetic transverse electric-transverse magnetic wave propagation”, J. Math. Phys., 54:4 (2013), 042902, 14 pp. | DOI | MR | Zbl

[11] Smirnov Yu. G., Valovik D. V., “Coupled electromagnetic transverse-electric-transverse magnetic wave propagation in a cylindrical waveguide with Kerr nonlinearity”, J. Math. Phys., 54:4 (2013), 043506, 22 pp. | DOI | MR | Zbl

[12] Smirnov Yu. G., Valovik D. V., “Problem of nonlinear coupled electromagnetic TE-TE wave propagation”, J. Math. Phys., 54:8 (2013), 083502, 13 pp. | DOI | MR | Zbl

[13] Stretton Dzh. A., Teoriya elektromagnetizma, GITTL, M.–L., 1948

[14] Landau L. D., Lifshits E. M., Teoreticheskaya fizika, V 10 t., v. VIII, Elektrodinamika sploshnykh sred, Fizmatlit, M., 2001

[15] Vainshtein L. A., Elektromagnitnye volny, Radio i svyaz, M., 1988

[16] Erugin N. P., Kniga dlya chteniya po obschemu kursu differentsialnykh uravnenii, Nauka i tekhn., Minsk, 1979 | MR | Zbl

[17] Abramov A. A., Dyshko A. L., Konyukhova N. B., Levitina T. V., “Vychislenie uglovykh volnovykh funktsii Lame resheniem vspomogatelnykh differentsialnykh uravnenii”, Zh. vychisl. matem. i matem. fiz., 29:6 (1989), 813–830 | MR

[18] Akhmediev N. N., Ankevich A., Solitony, Fizmatlit, M., 2003