Abdullaev, F. Kh.; Bouketir, A.; Messikh, A.; Umarov, B. A. Modulational instability and discrete breathers in the discrete cubic-quintic nonlinear Schrödinger equation. (English) Zbl 1125.37052 Physica D 232, No. 1, 54-61 (2007). Summary: We investigate the properties of modulational instability and discrete breathers in the cubic-quintic discrete nonlinear Schrödinger equation. We analyze regions of modulational instabilities of nonlinear plane waves and derive conditions for the existence and stability for bright discrete breather solutions. It is shown that the quintic nonlinearity brings qualitatively new conditions for stability of strongly localized modes. The application to the existence of localized modes in the Bose-Einstein condensate (BEC) with three-body interactions in an optical lattice is discussed. The numerical simulations agree with the analytical predictions. Cited in 14 Documents MSC: 37K60 Lattice dynamics; integrable lattice equations 35Q55 NLS equations (nonlinear Schrödinger equations) 37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems 37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) 81V70 Many-body theory; quantum Hall effect Keywords:modulational instability; discrete breather; discrete nonlinear Schrödinger equation; stability PDFBibTeX XMLCite \textit{F. Kh. Abdullaev} et al., Physica D 232, No. 1, 54--61 (2007; Zbl 1125.37052) Full Text: DOI References: [1] Page, J. B., Phys. Rev. B, 41, 7835 (1990) [2] Flash, S.; Willis, C. R., Phys. Rep., 295, 181 (1998) [3] The focus issue on “Nonlinear localized modes: Physics and applications”, Chaos, 13, 586 (2003) [4] Lederer, F.; Darmanyan, S.; Kobyakov, A., (Abdullaev, F. Kh.; Bang, O.; Soerensen, M. P., Nonlinearity and Disorder: Theory and Applications (2001), Kluwer AP), 131-157 [5] Sato, M.; Sievers, A., Nature, 432, 486 (2004) [6] Christodoulides, D. N.; Joseph, R. I., Opt. Lett., 13, 794 (1988) [7] Trombettoni, A.; Smerzi, A., Phys. Rev. Lett., 86, 2353 (2001) [8] Abdullaev, F. Kh.; Baizakov, B. B.; Darmanyan, S. A.; Konotop, V. V.; Salerno, M., Phys. Rev. A, 64, 043606 (2001) [9] Brazhnyi, V. A.; Konotop, V. V., Modern Phys. Lett. B, 18, 627 (2004) [10] Alfimov, G.; Kevrekidis, P. G.; Konotop, V. V.; Salerno, M., Phys. Rev. E, 66, 046608 (2002) [11] Trombettoni, A.; Kevrekidis, P. G.; Nistazakis, H. E.; Frantzeskakis, D. J., J. Phys. B, 39, S231 (2006) [12] Kevrekidis, P. G.; Frantzeskakis, D. J., Modern Phys. Lett. B, 18, 173 (2004) [13] Abdullaev, F. Kh.; Konotop, V. V., Phys. Rev. A, 68, 013605 (2003) [14] Morsh, O.; Oberthaler, M., Rev. Modern Phys., 78, 179 (2006) [15] Eiermann, B.; Anker, Th.; Ablietz, M.; Taglieber, M.; Treutlein, P.; Marzlin, K.-P.; Oberthaler, M. K., Phys. Rev. Lett., 92, 320401 (2004) [16] Zheng, W.; Wright, E. M.; Pu, H.; Meystre, P., Phys. Rev. A, 68, 023605 (2003) [17] Abdullaev, F. Kh.; Gammal, A.; Tomio, L.; Frederico, T., Phys. Rev. A, 63, 043604 (2001) [18] Akhmediev, N.; Das, M. P.; Vagov, A. V., Int. J. Mod. Phys. B, 13, 625 (1999) [19] Kraemer, T., Phys. Rev. Lett., 94, 123201 (2005) [20] Abdullaev, F. Kh.; Salerno, M., Phys. Rev. A, 72, 033617 (2005) [21] Alfimov, G.; Konotop, V. V.; Pacciani, P., Phys. Rev. A, 75, 023624 (2007) [22] Zhang, A. X.; Xue, J. K., Phys. Rev. A, 75, 013624 (2007) [23] Cataliotti, F., Science, 293, 843 (2001) [24] Sakaguchi, H.; Malomed, B. A., Phys. Rev. E, 72, 046610 (2005) [25] Abdullaev, F. Kh.; Abdumalikov, A. A.; Galimzyanov, R. M., Gap solitons in Bose-Einstein condensates in linear and nonlinear optical lattices, Phys. Lett. A (2007) [26] Stepic, M.; Kip, D.; Hadzievsi, L.; Maluckov, A., Phys. Rev. E, 69, 066618 (2004) [27] Kartashov, Y. V.; Vysloukh, A. V.; Torner, L., Opt. Express, 12, 2831 (2004) [28] De Angelis, C., IEEE J. Quantum Electron., 30, 818 (1994) [29] Carretero-Gonzales, R.; Talley, J. D.; Chong, C.; Malomed, B. A., Physica D, 216, 77 (2006) [30] Cai, D.; Bishop, A. R.; Gronblech-Jensen, N., Phys. Rev. Lett., 70, 591 (1993) [31] Abdullaev, F. Kh.; Darmanyan, S. A.; Garnier, J., (Wolf, E., Progress in Optics, vol. 42 (2002)), 301 [32] Darmanyan, S.; Kobyakov, A.; Lederer, F., J. Exp. Theor. Phys., 86, 682 (1998) [33] Smerzi, A.; Trombettoni, A., Phys. Rev. A, 68, 023613 (2003) [34] Kevrekidis, P. G.; Dmitriev, S. V.; Sukhorukov, A. A. [35] Khare, A.; Rasmussen, K.; Salerno, M.; Samuelsen, M.; Saxena, A., Phys. Rev. E, 74, 016607 (2006) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.