Stability of gap soliton complexes in the nonlinear Schrödinger equation with periodic potential and repulsive nonlinearity

P. P. Kizin

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 26 (2016) no. 4, pp. 591-602

Résumé

The work is devoted to numerical investigation of stability of stationary localized modes (“gap solitons”) for the one-dimentional nonlinear Schrödinger equation (NLSE) with periodic potential and repulsive nonlinearity. Two classes of the modes are considered: a bound state of a pair of in-phase and out-of-phase fundamental gap solitons (FGSs) from the first bandgap separated by various numbers of empty potential wells. Using the standard framework of linear stability analysis, we computed the linear spectra for the gap solitons by means of the Fourier collocation method and the Evans function method. We found that the gap solitons of the first and second classes are exponentially unstable for odd and even numbers of separating periods of the potential, respectively. The real parts of unstable eigenvalues in corresponding spectra decay exponentially with the distance between FGSs. On the contrary, we observed that the modes of the first and second classes are either linearly stable or exhibit weak oscillatory instabilities if the number of empty potential wells separating FGSs is even and odd, respectively. In both cases, the oscillatory instabilities arise in some vicinity of upper bandgap edge. In order to check the linear stability results, we fulfilled numerical simulations for the time-dependent NLSE by means of a finite-difference scheme. As a result, all the considered exponentially unstable solutions have been deformed to long-lived pulsating formations whereas stable solutions conserved their shapes for a long time.

Détail

Keywords: nonlinear Schrödinger equation, periodic potential, stability.
Mots-clés : gap solitons

Bibliographie

[1] Akylas T. R., Hwang G., Yang J., “From non-local gap solitary waves to bound states in periodic media”, Proc. R. Soc. A, 468 (2012), 116–135 | DOI | MR | Zbl

[2] Alfimov G. L., Avramenko A. I., “Coding of nonlinear states for the Gross–Pitaevskii equation with periodic potential”, Physica D: Nonlinear Phenomena, 254 (2013), 29–45 | DOI | MR | Zbl

[3] Alfimov G. L., Kevrekidis P. G., Konotop V. V., Salerno M., “Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential”, Physical Review E, 66 (2002), 046608, 6 pp. | DOI | MR

[4] Alfimov G. L., Kizin P. P., Zezyulin D. A., Gap solitons for the repulsive Gross–Pitaevskii equation with periodic potential: coding and method for computation, 2016, arXiv: 1609.00657 [nlin.PS]

[5] Alfimov G. L., Konotop V. V., Salerno M., “Matter solitons in Bose–Einstein condensates with optical lattices”, Europhysics Letters, 58 (2002), 7–13 | DOI

[6] Anderson M. H., Ensher J. R., Matthews M. R., Wieman C. E., Cornell E. A., “Observation of Bose–Einstein condensation in a dilute atomic vapor”, Science, 269 (1995), 198–201 | DOI

[7] Berezin F. A., Shubin M. A., The Schrödinger Equation, Kluwer, 1991 | DOI | MR | Zbl

[8] Bergé L., “Self-focusing dynamics of nonlinear waves in media with parabolic-type inhomogeneities”, Physics of Plasmas, 4 (1997), 1227 | DOI

[9] Blank E., Dohnal T., “Families of surface gap solitons and their stability via the numerical Evans function method”, SIAM J. Appl. Dyn. Syst., 10 (2011), 667–706 | DOI | MR | Zbl

[10] Bose S., “Planck's law and the light quantum hypothesis”, Journal of Astrophysics and Astronomy, 15 (1994), 3–7 | DOI

[11] Bradley C. C., Sackett C. A., Tollett J. J., Hulet R. G., “Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions”, Physical Review Letters, 75 (1995), 1687–1690 | DOI

[12] Chen H. H., Liu C. S., “Nonlinear wave and soliton propagation in media with arbitrary inhomogeneities”, The Physics of Fluids, 21 (1978), 377 | DOI | Zbl

[13] Dalfovo F., Giorgini S., Pitaevskii L. P., Stringari S., “Theory of Bose–Einstein condensation in trapped gases”, Reviews of Modern Physics, 71 (1999), 463–512 | DOI

[14] Davis K. B., Mewes M.-O., Andrews M. R., Druten N. J., Durfee D. S., Kurn D. M., Ketterle W., “Bose–Einstein condensation in a gas of sodium atoms”, Physical Review Letters, 75 (1995), 3969–3973 | DOI

[15] Edwards M., Burnett K., “Numerical solution of the nonlinear Schrödinger equation for small samples of trapped neutral atoms”, Physical Review A, 51 (1995), 1382–1386 | DOI

[16] Einstein A., Quantentheorie des einatomigen idealen Gases, Sitzungsber. Kgl. Preuss. Akad. Wiss., 1924

[17] Fukuizumi R., Sacchetti A., “Stationary states for nonlinear Schrödinger equations with periodic potentials”, Journal of Statistical Physics, 156 (2014), 707–738 | DOI | MR | Zbl

[18] Hwang G., Akylas T. R., Yang J., “Gap solitons and their linear stability in one-dimensional periodic media”, Physica D: Nonlinear Phenomena, 240 (2011), 1055–1068 | DOI | Zbl

[19] Kizin P. P., Zezyulin D. A., Alfimov G. L., “Oscillatory instabilities of gap solitons in a repulsive Bose–Einstein condensate”, Physica D: Nonlinear Phenomena, 337 (2016), 58–66 | DOI | MR

[20] Kumar S., Hasegawa A., “Quasi-soliton propagation in dispersion-managed optical fibers”, Optics Letters, 22 (1997), 372–374 | DOI

[21] Kunze M., Küpper T., Mezentsev V. K., Shapiro E. G., Turitsyn S., “Nonlinear solitary waves with Gaussian tails”, Physica D: Nonlinear Phenomena, 128 (1999), 273–295 | DOI | MR | Zbl

[22] Louis P. J. Y., Ostrovskaya E. A., Savage C. M., Kivshar Yu. S., “Bose–Einstein condensates in optical lattices: Band-gap structure and solitons”, Physical Review A, 67 (2003), 013602, 9 pp. | DOI

[23] Pelinovsky D. E., Kevrekidis P. G., Frantzeskakis D. J., “Stability of discrete solitons in nonlinear Schrödinger lattices”, Physica D: Nonlinear Phenomena, 212 (2005), 1–19 | DOI | MR | Zbl

[24] Pelinovsky D. E., Sukhorukov A. A., Kivshar Yu. S., “Bifurcations and stability of gap solitons in periodic potentials”, Physical Review E, 70 (2004), 036618, 17 pp. | DOI | MR

[25] Trofimov V. A., Peskov N. V., “Comparison of finite-defference schemes for the Gross–Pitaevskii equation”, Mathematical Modelling and Analysis, 14 (2009), 109–126 | DOI | MR | Zbl

[26] Turitsyn S. K., “Stability of an optical soliton with Gaussian tails”, Physical Review E, 56 (1997), R3784–R3787 | DOI

[27] Turitsyn S. K., “Theory of average pulse propagation in high-bit-rate optical transmission systems with strong dispersion management”, Journal of Experimental and Theoretical Physics Letters, 65 (1997), 845–851 | DOI

[28] Wang J., Yang J., Alexander T. J., Kivshar Yu. S., “Truncated-Bloch-wave solitons in optical lattices”, Physical Review A, 79 (2009), 043610, 5 pp. | DOI

[29] Yang J., Nonlinear waves in integrable and nonintegrable systems, SIAM, 2010 | MR | Zbl

[30] Zhang Y., Wu B., “Composition relation between gap solitons and Bloch waves in nonlinear periodic systems”, Physical Review Letters, 102 (2009), 093905, 4 pp. | DOI