Nonlocal nonlinear Schrödinger equation and its discrete version: soliton solutions and gauge equivalence. (English) Zbl 1352.35163
Searching for solutions of the soliton equations and discovering the relations between different nonlinear evolution equations are important work of studying soliton equations. Many systematic methods have been developed to obtain explicit solutions of soliton equations, among which the Darboux transformation is the most famous one. In this paper, the authors first discussed gauge equivalence of the nonlocal focusing/defocusing NLS equation and their discrete cases. Then, depending on the Darboux transformation, soliton solutions of the discrete nonlocal NLS equations (including the focusing and defocusing cases). As an application, some interesting figures are plotted. The research of the nonlocal NLS equation and the discrete case in this paper has discovered many new properties of these equations.
Reviewer: Xue Bo (Zhengzhou)
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 39A12 | Discrete version of topics in analysis |
| 35C08 | Soliton solutions |
| 70S15 | Yang-Mills and other gauge theories in mechanics of particles and systems |
| 35Q51 | Soliton equations |
| 37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
Keywords:
nonlocal NLS equation; discrete; gauge equivalence; Darboux transformation; soliton solutionsReferences:
| [1] | Ablowitz, M. J.; Musslimani, Z. H., Phys. Rev. Lett., 110, 064105 (2013) · doi:10.1103/PhysRevLett.110.064105 |
| [2] | Bender, C. M.; Boettcher, S., Phys. Rev. Lett., 80, 5243 (1998) · Zbl 0947.81018 · doi:10.1103/PhysRevLett.80.5243 |
| [3] | Rüter, C. E.; Makris, K. G.; El-Ganainy, R.; Christodoulides, D. N.; Segev, M.; Kip, D., Nat. Phys., 6, 192 (2010) · doi:10.1038/nphys1515 |
| [4] | Guo, A.; Salamo, G. J.; Duchesne, D.; Morandotti, R.; Volatier-Ravat, M.; Aimez, V.; Siviloglou, G. A.; Christodoulides, D. N., Phys. Rev. Lett., 103, 093902 (2009) · doi:10.1103/PhysRevLett.103.093902 |
| [5] | Regensburger, A.; Bersch, C.; Miri, M.-A.; Onishchukov, G.; Christodoulides, D. N., Nature, 488, 167-171 (2012) · doi:10.1038/nature11298 |
| [6] | Sarma, A. K.; Miri, M. A.; Musslimani, Z. H.; Christodoulides, D. N., Phys. Rev. E, 89, 052918 (2014) · doi:10.1103/PhysRevE.89.052918 |
| [7] | Valchev, T.; Slavova, A., Mathematics in Industry, 36-52 (2014) |
| [8] | Li, M.; Xu, T., Phys. Rev. E, 91, 033202 (2015) · doi:10.1103/PhysRevE.91.033202 |
| [9] | Ablowitz, M. J.; Musslimani, Z. H., Phys. Rev. E, 90, 042912 (2014) · doi:10.1103/PhysRevE.90.032912 |
| [10] | Lakshmanan, M., Phys. Lett. A., 61, 53 (1977) · doi:10.1016/0375-9601(77)90262-6 |
| [11] | Zakharov, V. E.; Takhtajan, L. A., Theor. Math. Phys., 38, 17 (1979) · doi:10.1007/BF01030253 |
| [12] | Kundu, A., J. Math. Phys., 25, 3433 (1984) · doi:10.1063/1.526113 |
| [13] | Kundu, A., J. Phys. A: Math. Gen., 19, 1303 (1986) · Zbl 0654.35084 · doi:10.1088/0305-4470/19/8/012 |
| [14] | Ding, Q., Phys. Lett. A, 248, 49 (1998) · Zbl 1115.35368 · doi:10.1016/S0375-9601(98)00697-5 |
| [15] | Ishimori, Y., J. Phys. Soc. Jpn., 52, 3417 (1982) · doi:10.1143/JPSJ.51.3417 |
| [16] | Ding, Q., Phys. Lett. A, 266, 146 (2000) · Zbl 0949.37048 · doi:10.1016/S0375-9601(00)00027-X |
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.
