Geodesic
The Riemann–Hilbert Problem for Analytic Description of the DM Solitons
Teoretičeskaâ i matematičeskaâ fizika, Tome 137 (2003) no. 3, pp. 433-444 Cet article a éte moissonné depuis la source Math-Net.Ru

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A simple exact formula is derived for the profile of the optical pulse propagating over a DM fiber with zero mean dispersion. The dissipation is neglected, and dispersion is assumed to be constant along the adjacent legs of the waveguide, thus providing the applicability of the integrable NLS models within each leg. The formula describes a class of solutions called dispersion-managed solitons (DM solitons), which are periodic along the waveguide and exponentially localized in time. The DM solitons are parameterized by a certain class of spectral data, specified from numerical simulations. Using a related Riemann–Hilbert problem, we reconstruct a profile of the DM soliton from the given spectral data. For sufficiently long legs, the leading term of DM soliton is found in explicit form by asymptotic undressing of the Riemann–Hilbert problem. The analytic results are compared with numerical simulations.
Mots-clés : solitons, dispersion management
Keywords: nonlinear Schrödinger equation with periodic dispersion, Riemann–Hilbert problem, inverse scattering transform. 
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A. V. Mikhailov; V. Yu. Novokshenov. The Riemann–Hilbert Problem for Analytic Description of the DM Solitons. Teoretičeskaâ i matematičeskaâ fizika, Tome 137 (2003) no. 3, pp. 433-444. http://geodesic.mathdoc.fr/item/TMF_2003_137_3_a9/

[1] V. E. Zakharov, S. Wabnitz (eds.), Optical Solitons: Theoretical Challenges and Industrial Perspectives, Springer, Berlin, 1999

[2] I. R. Gabitov, S. K. Turitsin, Opt. Lett., 21 (1996), 327 ; В. Е. Захаров, С. В. Манаков, Письма в ЖЭТФ, 70 (1999), 573; С. К. Турицын, В. К. Мезенцев, Письма в ЖЭТФ, 67 (1998), 616 | DOI

[3] J. H. Nijhof, W. Forysiak, N. J. Doran, IERE J. Quantum Electr., 6 (2000), 330 ; D. E. Pelinovsky, V. Zharnitsky, SIAM J. Appl. Math., 63 (2003), 745 ; V. Zharnitsky, E. Grenier, C. K. R. T. Jones, S. Turitsyn, Physica D, 152–153 (2001), 794 ; T. Yang, W. L. Kath, Physica D, 149 (2001), 80 | DOI | DOI | MR | Zbl | DOI | MR | Zbl | DOI | Zbl

[4] M. J. Ablowitz, G. Biondini, Opt. Lett., 23 (1998), 1668 | DOI

[5] P. M. Lushnikov, Opt. Lett., 26 (2001), 1535 | DOI

[6] T. Hirooka, A. Hasegawa, Opt. Lett., 23 (1998), 768 | DOI

[7] V. E. Zakharov, A. B. Shabat, ZhETF, 61 (1971), 118

[8] V. E. Zakharov, S. V. Manakov, ZhETF, 71 (1976), 203 ; В. Ю. Новокшенов, ДАН СССР, 251 (1980), 799 | Zbl

[9] J. H. B. Nijhof, S. K. Turitsyn, N. J. Doran, “Dispersion-managed solitons and the inverse scattering transform”, Nonlinear Guided Waves and Their Applications, OSA Technical Digest, OSA, Washington, 1999, 313

[10] P. A. Deift, A. R. Its, X. Zhou, Ann. Math., 146 (1997), 149 | DOI | MR | Zbl

[11] F. Göhmann, A. G. Izergin, V. E. Korepin, G. G. Pronko, Int. J. Mod. Phys. B, 12 (1998), 2409 | DOI

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

[13] A. V. Mikhailov, V. Yu. Novokshenov, Pisma v ZhETF, 73 (2001), 287

[14] N. M. Bogolyubov, A. G. Izergin, V. E. Korepin, Korrelyatsionnye funktsii integriruemykh sistem i kvantovyi metod obratnoi zadachi, Nauka, M., 1992 | MR | Zbl