Geodesic
Multimode systems of nonlinear equations: Derivation, integrability,
Teoretičeskaâ i matematičeskaâ fizika, Tome 168 (2011) no. 1, pp. 138-150 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider the propagation of electromagnetic pulses in isotropic media taking a third-order nonlinearity into account. We develop a method for transforming Maxwell's equations based on a complete set of projection operators corresponding to wave-dispersion branches (in a waveguide or in matter) with the propagation direction taken into account. The most important result of applying the method is a system of equations describing the one-dimensional dynamics of pulses propagating in opposite directions without accounting for dispersion. We derive the corresponding self-action equations. We thus introduce dispersion in the media and show how the operators change. We obtain generalized Schäfer–Wayne short-pulse equations accounting for both propagation directions. In the three-dimensional problem, we focus on optic fibers with dispersive matter, deriving and numerically solving equations of the waveguide-mode interaction. We discuss the effects of the interaction of unidirectional pulses. For the coupled nonlinear Schrödinger equations, we discuss a concept of numerical integrability and apply the developed calculation schemes.
Keywords: projection-operator method, multimode waveguide, coupled nonlinear Schrödinger equations, short-pulse equation. 
@article{TMF_2011_168_1_a10,
     author = {M. Kuszner and S. B. Leble and B. Reichel},
     title = {Multimode systems of nonlinear equations: {Derivation,} integrability,},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {138--150},
     year = {2011},
     volume = {168},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2011_168_1_a10/}
}
TY  - JOUR
AU  - M. Kuszner
AU  - S. B. Leble
AU  - B. Reichel
TI  - Multimode systems of nonlinear equations: Derivation, integrability,
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2011
SP  - 138
EP  - 150
VL  - 168
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2011_168_1_a10/
LA  - ru
ID  - TMF_2011_168_1_a10
ER  - 
%0 Journal Article
%A M. Kuszner
%A S. B. Leble
%A B. Reichel
%T Multimode systems of nonlinear equations: Derivation, integrability,
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2011
%P 138-150
%V 168
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2011_168_1_a10/
%G ru
%F TMF_2011_168_1_a10
M. Kuszner; S. B. Leble; B. Reichel. Multimode systems of nonlinear equations: Derivation, integrability,. Teoretičeskaâ i matematičeskaâ fizika, Tome 168 (2011) no. 1, pp. 138-150. http://geodesic.mathdoc.fr/item/TMF_2011_168_1_a10/

[1] M. Kolesik, J. V. Moloney, M. Mlejnek, Phys. Rev. Lett., 89:28 (2002), 283902, 4 pp. | DOI

[2] P. Kinsler, S. B. P. Radnor, G. H. C. New, Phys. Rev. A, 72:6 (2005), 063807, arXiv: physics/0611215 | DOI

[3] P. Kinsler, J. Opt. Soc. Amer. B, 24:9 (2007), 2363–2368, arXiv: 0707.0986 | DOI

[4] P. Kinsler, Phys. Rev. A, 81:2 (2010), 023808, 8 pp., arXiv: 0909.3407 | DOI

[5] S. B. Leble, Nonlinear Waves in Waveguides, Springer, Berlin, 1991

[6] T. Schäfer, C. E. Wayne, Propagation of Ultra-short Optical Pulses in Nonlinear Media, Elsevier, New York, 2002

[7] Y. Chung, C. K. R. T. Jones, T. Schäfer, C. E. Wayne, Nonlinearity, 18:3 (2005), 1351–1374, arXiv: nlin/0408020 | DOI | MR | Zbl

[8] S. Sakovich, J. Phys. Soc. Jpn., 77 (2008), 123001, 4 pp., arXiv: 0801.3179 | DOI

[9] A. Sakovich, S. Sakovich, J. Phys. A, 39:22 (2006), L361–L367, arXiv: nlin/0601019 | DOI | MR | Zbl

[10] S. Leble, “Nonlinear waves in optical waveguides and soliton theory applications”, Optical solitons. Theoretical and Experimental Challenges, Lecture Notes in Physics, 613, eds. K. Porsezian, V. C. Kuriakose, Springer, 2003, 71–104 | DOI

[11] S. B. Leble, B. Reichel, J. Mod. Opt., 55:1 (2008), 1–11 | DOI | MR | Zbl

[12] S. Leble, B. Reichel, Eur. Phys. J., 173:1 (2009), 5–55 | DOI

[13] V. E. Zakharov, PMTF, 2 (1968), 86–94 | DOI

[14] A. L. Berkhoer, V. E. Zakharov, ZhETF, 58:3 (1970), 903–911

[15] B. Reichel, S. Leble, Comput. Math. Appl., 55:4 (2008), 745–759, arXiv: math/0611420 | DOI | MR | Zbl

[16] S. Kelikh, Molekulyarnaya nelineinaya optika, M., Nauka, 1981

[17] G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, New York, 2001

[18] A. Perelomova, Phys. Lett. A, 357:1 (2006), 42–47 | DOI | Zbl

[19] A. Hasegawa, Y. Kodama, Proc. IEEE, 69:9 (1981), 1145–1150 | DOI

[20] A. Hasegawa, Y. Kodama, Solitons in Optical Communication, Clarendon Press, Oxford, 1995 | Zbl

[21] S. B. Leble, B. Reichel, J. Modern Opt., 55:21 (2008), 3651–3664 | DOI | MR

[22] M. Pietrzyk, I. Kanattšikow, Multisymplectic analysis of the short pulse equation and numerical application, WIAS Preprint No 1278, 2007\par {http://www.wias-berlin.de/preprint/1278/wias_preprints_1278.pdf}