Geodesic
Conservative finite-difference scheme for the problem of a femtosecond laser pulse with an axially symmetric profile propagating in a medium with cubic nonlinearity
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 10, pp. 1752-1773 Cet article a éte moissonné depuis la source Math-Net.Ru

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Conservative finite-difference schemes are constructed for the problem of a femtosecond laser pulse propagating in a cubically nonlinear medium in the axially symmetric case with allowance for temporal dispersion of the nonlinear response of the medium. The process is governed by the nonlinear Schrödinger equation involving the time derivative of the nonlinear term. The invariants of the differential problem are presented. It is shown that the difference analogues of these invariants hold for the solution to the finite-difference schemes proposed for the problem. As an example, the numerical results obtained for the self-focusing of a femtosecond light beam are presented. 
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A. G. Volkov; V. A. Trofimov. Conservative finite-difference scheme for the problem of a femtosecond laser pulse with an axially symmetric profile propagating in a medium with cubic nonlinearity. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 10, pp. 1752-1773. http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_10_a8/

[1] Akhmanov C. A., Vysloukh V. A., Chirkin A. C., Optika femtosekundnykh lazernykh impulsov, Nauka, M., 1988

[2] Sukhorukov A. P., Nelineinye volnovye vzaimodeistviya v optike i radiofizike, Nauka, M., 1988

[3] Borovskii A. B., Galkin A. L., Lazernaya fizika, Izd-vo AT, M., 1996

[4] Koroteev N. I., Shumai I. L., Fizika moschnogo lazernogo izlucheniya, Nauka, M., 1991

[5] Agraval G., Nelineinaya volokonnaya optika, Mir, M., 1996

[6] Varentsova S. A., Volkov A. G., Trofimov B. A., “Konservativnaya raznostnaya skhema dlya zadachi rasprostraneniya femtosekundnogo lazernogo impulsa v kubichno-nelineinoi srede”, Zh. vychisl. matem. i matem. fiz., 43:11 (2003), 1709–1721 | MR | Zbl

[7] Trofimov V. A., “K voprosu ob invariantakh nelineinogo rasprostraneniya femtosekundnykh impulsov”, Izv. vuzov. Radiofizika, 35:6–7 (1992), 618–621

[8] Trofimov V. A., “Invarianty nelineinogo rasprostraneniya femtosekundnykh impulsov”, Izv. vuzov. Radiofizika, 62:4 (1999), 369–372 | MR

[9] Trofimov V. A., “O novom podkhode k modelirovaniyu nelineinogo rasprostraneniya sverkhkorotkikh lazernykh impulsov”, Zh. vychisl. matem. i matem. fiz., 38:5 (1998), 835–839 | MR | Zbl

[10] Volkov A. G., Trofimov V. A., Tereshin E. B., “Konservativnye raznostnye skhemy dlya nekotorykh zadach femtosekundnoi optiki”, Differents. ur-niya, 41:7 (2005), 908–917 | MR | Zbl

[11] Askaryan G. A., “Vozdeistvie gradienta polya intensivnogo elektromagnitnogo lucha na elektrony i atomy”, Zh. eksperim. i teor. fiz., 42:6 (1962), 1567–1570

[12] Akhmanov C. A., Sukhorukov A. P., Khokhlov R. V., “Samofokusirovka i difraktsiya sveta v nelineinoi srede”, Uspekhi fiz. nauk, 93:1 (1967), 19–70

[13] Lugovoi V. N., Prokhorov A. M., “Teoriya rasprostraneniya moschnogo lazernogo izlucheniya v nelineinoi srede”, Uspekhi fiz. nauk, 111:2 (1973), 203–260

[14] Berge L., “Wave collapse in physics: principles and applications to light and plasma waves”, Phys. Repts., 303:5–6 (1998), 259–370 | DOI | MR

[15] Feit M. D., Fleck. J. A., “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams”, J. Optimizat. Soc. Amer. B, 5:3 (1988), 633–640 | DOI

[16] Fibich G., Malkin V. M., Papanicolaou G. C., “Beam self-focusing in the presence of small normal time dispersion”, Phys. Rev. A, 52:5 (1995), 4218–4228 | DOI

[17] Chernev P., Petrov V., “Self-focusine of lieht pulses in the presence of normal group-velocity dispersion”, Optimizat. Letts, 17:3 (1992), 172–174 | DOI

[18] Luther G. G., Moloney J. V., Newell A. C., Wright E. M., “Self-focusing threshold is normally dispersive media”, Optimizat. Letts, 19:12 (1994), 862–864 | DOI

[19] Kunitsyn S. D., Sukhorukov A. P., Trofimov B. A., “Samovozdeistvie opticheskogo izlucheniya v usloviyakh generatsii vstrechnoi volny”, Izv. RAN. Ser. Fiz., 56:12 (1992), 201–208 | MR

[20] Silbergberg Y., “Collapse of optical beams”, Optimizat. Letts, 15:22 (1990), 1282–1284 | DOI

[21] Kivshar Yu. S., Pelinovsky D. E., “Self-focusing and transverse instabilities of solitary waves”, Phys. Repts, 331:4 (2000), 117–195 | DOI | MR

[22] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1989 | MR

[23] Samarskii A. A., Andreev V. B., Raznostnye metody dlya ellipticheskikh uravnenii, Nauka, M., 1976 | MR | Zbl

[24] Samarskii A. A., Nikolaev E. C., Metody resheniya setochnykh uravnenii, Nauka, M., 1978 | MR