Iterative method for finding the eigenfunctions of a system of two Schrödinger equations with combined nonlinearity

O. V. Matusevich   ; V. A. Trofimov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 4, pp. 713-724

Résumé

An iterative method is proposed to determine the eigenfunctions of a system of two nonlinear Schrödinger equations governing the interaction of two femtosecond pulses in a medium with quadratic and cubic nonlinearity. The method produces soliton solutions of a new form for a wide range of nonlinearity coefficients corresponding to the first and second eigenvalues. A specially chosen initial approximation is required to determine the third and higher eigenfunctions.

Bibliographie

[1] Zakharov V. E., Manakov C. B., Novikov S. P., Pitaevskii L. P., Teoriya solitonov: Metod obratnoi zadachi, Nauka, M., 1980 | MR

[2] Kerner B. C., Osipov V. V., Avtosolitony, Nauka, M., 1991 | Zbl

[3] Abdullaev F., Darmanyan S., Khabibullaev P., Optical solitons, Berlin etc., 1993

[4] Ablovits M., Sigur X., Solitony i metod obratnoi zadachi, Mir, M., 1987 | MR

[5] Ablowitz M. J., Clarkson P. A., Solitons. Nonlinear evolution equations and inverse scattering, London Math. Soc. Lect. Notes, 149, Cambridge Univ. Press, Cambridge, 1991 | MR | Zbl

[6] Takhtadzhyan L. A., Faddeev L. D., Gamiltonov podkhod v teorii solitonov, Nauka, M., 1986 | MR | Zbl

[7] Lem Dzh. L., Vvedenie v teoriyu solitonov, Mir, M., 1983 | MR

[8] Bogoyavlenskii O. I., Oprokidyvayuschie solitony. Nelineinye integriruemye uravneniya, Nauka, M., 1991 | MR | Zbl

[9] Nayanov V. I., Mnogopolevye solitony, Fizmatlit, M., 2006 | MR | Zbl

[10] Dodd R., Eilbek Dzh., Gibbon Dzh. i dr., Solitony i nelineinye volnovye uravneniya, Mir, M., 1988 | MR

[11] Miva T.,Dzhimbo M.,Date E., Solitony: differentsialnye uravneniya, simmetrii i beskonechnomernye algebry, MTsNMO, M., 2005

[12] Infeld E., Roulands Dzh., Nelineinye volny, solitony i khaos, Fizmatlit, M., 2005

[13] Nyuell A., Solitony v matematike i fizike, Mir, M., 1989 | MR

[14] Kalodzhero F., Degasperis A., Spektralnye preobrazovaniya i solitony. Metody resheniya i issledovaniya nelineinykh evolyutsionnykh uravnenii, Mir, M., 1985 | MR

[15] Davydov A. C., Solitony v molekulyarnykh sistemakh, Nauk. dumka, Kiev, 1984 | MR

[16] Novokshenov V. Yu., Vvedenie v teoriyu solitonov, RKhD, Moskva, Izhevsk, 2002

[17] Radzharaman R., Solitony i instantony v kvantovoi teorii polya, Mir, M., 1985

[18] Kivshar Yu. S., Agraval G. P., Opticheskie solitony. Ot svetovodov k fotonnym kristallam, Fizmatlit, M., 2005

[19] Akhmediev H. H., Ankevich A., Solitony. Nelineinye impulsy i puchki, Fizmatlit, M., 2003

[20] Karamzin Yu. N., Sukhorukov A. P., “Nelineinoe vzaimodeistvie difragiruyuschikh svetovykh puchkov v srede s kvadratichnoi nelineinostyu, vzaimofokusirovka puchkov i ogranichenie effektivnosti opticheskikh preobrazovatelei chastoty”, Pisma v ZhETF, 20:11 (1974), 734–739

[21] Buryak A. V., Kivshar Yu. S., “Spatial optical solitons governed by quadratic nonlinearity”, Optimizat. Letts, 19:20 (1994), 1612–1615 | DOI

[22] Buryak A. V., Trapani P. D., Skryabin D. V., Trillo S., “Optical solitons due to quadratic nonlinearities: From basic physics to futuristic applications”, Phys. Rept., 370:2 (2002), 63–235 | DOI | MR | Zbl

[23] Brail L., Lange H., “Stationary, oscillatory and solitary wave type solution of singular nonlinear Schrödinger equations”, Math. Meth. Appl. Sci., 8:4 (1986), 559–575 | DOI | MR

[24] Liu X., Beckwitt K., Wise F. W., “Two-dimensional optical spatiotemporal solitons in quadratic media”, Phys. Rev. E, 62:1 (2000), 1328–1340 | DOI

[25] Stegeman G., Hagau D. J., Tomer L., “$\chi^{(2)}$ cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons”, Optimizat. Quantum Electron., 28:12 (1996), 1691–1740 | DOI

[26] Ashihara S., Nishina J., Shimura T., Kuroda K., “Soliton compression of femtosecond pulses in quadratic media”, JOSA B, 19:10 (2002), 2505–2510 | DOI

[27] Grimshaw R., Kuznetsov E. A., Shapiro E. G., The two-parameter soliton family for the inter, Fizmatlit, M., 2003 | Zbl

[28] Malomed B. A., Mihalache D., Wise F., Torner L., “Spatiotemporal optical solitons”, J. Optimizat. B, 7:5 (2005), 53–72

[29] Demmel Dzh., Vychislitelnaya lineinaya algebra. Teoriya i prilozheniya, Mir, M., 2001

[30] Golub Dzh., Van Loun Ch., Matrichnye vychisleniya, Mir, M., 1999

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

[32] Varentsova S. A., Trofimov V. A., “O raznostnom metode nakhozhdeniya sobstvennykh mod nelineinogo uravneniya Shredingera”, Vestn. MGU. Ser. 15, 2005, no. 3, 16–22 | MR | Zbl

[33] Trofimov V. A., Varentsova S. A., “Computational method for finding of soliton solutions of a nonlinear Schrödinger equation”, Lect. Notes Math., 3401, Springer, Berlin etc., 2005, 550–557

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

[35] Shen I. R., Printsipy nelineinoi optiki, Nauka, M., 1989

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

[37] Samarskii A. A., Gulin A. B., Chislennye metody, Nauka, M., 1989 | MR

[38] Trofimov V. A., Matusevich O. V., “Bistable regime of SHG of femtosecond laser pulse in optical fiber”, Photonics Applic. Industry and Res., v. IV, Proc. SPIE, 5949, 2005, 300–306

[39] Di Trapani P., Chinaglia W., Minardi S., “Observation of quadratic optical vortex soliton”, Phys. Rev. Letts, 84:17 (2000), 3843–3846 | DOI