Solitary wave solutions as a signature of the instability in the discrete nonlinear Schrödinger equation. (English) Zbl 1233.35177
Summary: The effect of instability on the propagation of solitary waves along one-dimensional discrete nonlinear Schrödinger equation with cubic nonlinearity is revisited. A self-contained quasicontinuum approximation is developed to derive closed-form expressions for small-amplitude solitary waves. The notion that the existence of nonlinear solitary waves in discrete systems is a signature of the modulation instability is used. With the help of this notion we conjecture that instability effects on moving solitons can be qualitative estimated from the analytical solutions. Results from numerical simulations are presented to support this conjecture.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35C08 | Soliton solutions |
| 39A12 | Discrete version of topics in analysis |
| 82D20 | Statistical mechanics of solids |
| 37M05 | Simulation of dynamical systems |
References:
| [1] | Vicencio, R. A.; Brand, J.; Flach, S., Phys. Rev. Lett., 98, 184102 (2007) |
| [2] | Schumm, T.; Hofferberth, S.; Andersson, L. M.; Wildermuth, S.; Groth, S.; Bar-Joseph, I.; Schmiedmayer, J.; Krüger, P., Nature Phys., 1, 57 (2005) |
| [3] | Kay, A.; Pachos, J. K.; Adams, C. S., Phys. Rev. A, 73, 022310 (2006) |
| [4] | Ginsberg, N. S.; Garner, S. R.; Hau, L. V., Nature, 445, 623 (2007) |
| [5] | Oxtoby, O. F.; Barashenkov, I. V., Phys. Rev. E., 76, 036603 (2007) |
| [6] | Vakhnenko, A. A.; Gaididei, Yu. B., Theor. Math. Phys., 68, 873 (1986) |
| [7] | Kivshar, Yu. S.; Campbell, D. K., Phys. Rev. E, 48, 3077 (1993) |
| [8] | Claude, Ch.; Kivshar, Yu. S.; Kluth, O.; Spatschek, K. H., Phys. Rev. B, 47, 14228 (1993) |
| [9] | Aceves, A. B.; De Angelis, C.; Peschel, T.; Muschall, R.; Lederer, F.; Trillo, S.; Wabnitz, S., Phys. Rev. E, 53, 1172 (1996) |
| [10] | Gómez-Gardañes, J., L.M. Floría, M. Peyrard, and A. R. Bishop, Chaos, 14, 1130 (2004) · Zbl 1080.35139 |
| [11] | Ablowitz, M. J.; Musslimani, Z. H.; Biondini, G., Phys. Rev. E, 65, 026602 (2002) |
| [12] | Peyrard, M., Nonlinearity, 17, R1 (2004) · Zbl 1092.82015 |
| [13] | Trombettoni, A.; Smerzi, A., Phys. Rev. Lett., 86, 2353 (2001) |
| [14] | Pelinovsky, D.; Rothos, V. M., Physica D, 202, 16 (2005) · Zbl 1086.34059 |
| [15] | Kivshar, Y. S.; Królikowski, W.; Chubykalo, O. A., Phys. Rev. E, 50, 5020 (1994) |
| [16] | Johansson, M.; Kivshar, Y. S., Phys. Rev. Lett., 82, 85 (1999) |
| [17] | Sanchéz-Rey, B.; Johansson, M., Phys. Rev. E, 71, 036627 (2005) |
| [18] | Cataliotti, F. S.; Burger, S.; Fort, C.; Maddaloni, P.; Minardi, F.; Trombettoni, A.; Smerzi, A.; Inguscio, M., Science, 293, 843 (2001) |
| [19] | Smerzi, A.; Trombettoni, A., Phys. Rev. A, 68, 023613 (2003) |
| [20] | Alfimov, G. L.; Kevrekidis, P. G.; Konotop, V. V.; Salerno, M., Phys. Rev. E, 66, 046608 (2002) |
| [21] | Carretero-González, R.; Frantzeskakis, D. J.; Kevrekidis, P. G., Nonlinearity, 21, R139 (2008) · Zbl 1216.82023 |
| [22] | Kivshar, Y. S.; Peyrard, M., Phys. Rev. A, 46, 3198 (1992) |
| [23] | Daumont, I.; Dauxoix, T.; Peyrard, M., Nonlinearity, 10, 617 (1997) · Zbl 0906.76027 |
| [24] | Konotop, V. V.; Salerno, M., Phys. Rev. A, 65 (2002), 021602(R) |
| [25] | Lederer, F.; Stegeman, G. I.; Christodoulides, D. N.; Assanto, G.; Segev, M.; Silberberg, Y., Phys. Rep., 463, 1 (2008) |
| [26] | Pelinovsky, D. E.; Kevrekidis, P. G.; Frantzeskakis, D. J., Physica D, 212, 1 (2005) · Zbl 1088.39004 |
| [27] | Neuper, A.; Mertens, F. G.; Flytzanis, N., Z. Phys. B, 95, 397 (1995) |
| [28] | Brunhuber, C.; Mertens, F. G.; Gaididei, Y., Phys. Rev. E, 73, 016614 (2006) |
| [29] | Arévalo, E., Phys. Rev. E, 76, 066602 (2007) |
| [30] | Arévalo, E., Europhys. Lett., 83, 10004 (2008) |
| [31] | Arévalo, E., Phys. Rev. Lett., 102, 224102 (2009) |
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.
