Multi-soliton management by the integrable nonautonomous nonlinear integro-differential Schrödinger equation. (English) Zbl 1344.37079
Summary: We consider a wide class of integrable nonautonomous nonlinear integro-differential Schrödinger equation which contains the models for the soliton management in Bose-Einstein condensates, nonlinear optics, and inhomogeneous Heisenberg spin chain. With the help of the nonisospectral AKNS hierarchy, we obtain the N-fold Darboux transformation and the N-fold soliton-like solutions for the equation. The soliton management, especially the synchronized dispersive and nonlinear management in optical fibers is discussed. It is found that in the situation without external potential, the synchronized dispersive and nonlinear management can keep the integrability of the nonlinear Schrödinger equation; this suggests that in optical fibers, the synchronized dispersive and nonlinear management can control and maintain the propagation of a multi-soliton.
MSC:
| 37K40 | Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 37C60 | Nonautonomous smooth dynamical systems |
| 45K05 | Integro-partial differential equations |
| 37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
Keywords:
soliton management; nonisospectral AKNS hierarchy; N-fold Darboux transformation; nonautonomous nonlinear Schrödinger equationReferences:
| [1] | Malomed, B. A., Soliton Management in Periodic Systems (2006), Springer · Zbl 1214.35056 |
| [2] | Chen, H. H.; Liu, C. S., Phys. Rev. Lett., 37, 693-697 (1976) |
| [3] | Hasegawa, A.; Kodama, Y., Solitons in Optical Communications (1995), Oxford University Press: Oxford University Press Oxford · Zbl 0840.35092 |
| [4] | Nakazawa, M.; Kubota, H.; Suzuki, K.; Yamada, E.; Sahara, A., Chaos, 10, 486-514 (2000) |
| [5] | Mollenauer, L. F.; Gordon, J. P., Solitons in Optical Fibers (2006), Academic Press: Academic Press Boston |
| [6] | Turitsyn, S. K.; Aceves, A. B.; Jones, C. K.R. T.; Zharnitsky, V., Phys. Rev. E, 58, R48-R51 (1998) |
| [7] | Ganapathy, R., Commun. Nonlinear Sci. Numer. Simul., 17, 4544-4550 (2012) · Zbl 1264.35090 |
| [8] | Mani Rajan, M. S.; Mahalingamb, A.; Uthayakumarc, A., Ann. Phys., 346, 1-13 (2014) · Zbl 1342.35347 |
| [9] | Kevrekidis, P. G.; Theocharis, G.; Frantzeskakis, D. J.; Malomed, B. A., Phys. Rev. Lett., 90, 230401 (2003) |
| [10] | Serkin, V. N.; Hasegawa, A.; Belyaeva, T. L., Phys. Rev. Lett., 98, 074102 (2007) |
| [11] | Serkin, V. N.; Hasegawa, A.; Belyaeva, T. L., Phys. Rev. A, 81, 023610 (2010) |
| [12] | Calogero, F.; Degasperis, A., Lett. Nuovo Cimento, 22, 420-424 (1978) |
| [13] | Lakshmanan, M.; Bullough, R. K., Phys. Lett., 80, 287-292 (1980) |
| [14] | Balakrishnan, R., J. Phys. C: Solid State Phys., 15, L1305 (1982) |
| [15] | Balakrishnan, R., Phys. Rev. A, 32, 1144-1149 (1985) |
| [16] | Lakshmanan, M., Phil. Trans. R. Soc. A, 369, 1280-1300 (2011) · Zbl 1219.82139 |
| [17] | Rajendran, S.; Muruganandam, P.; Lakshmanan, M., J. Math. Phys., 52, 023515 (2011) · Zbl 1314.35171 |
| [18] | Atre, R.; Panigrahi, P. K.; Agarwal, G. S., Phys. Rev. E, 73, 056611 (2006) |
| [19] | He, X. G.; Zhao, D.; Li, L.; Luo, H.-G., Phys. Rev. E, 79, 056610 (2009) |
| [20] | Luo, H. G.; Zhao, D.; He, X.-G., Phys. Rev. A, 79, 063802 (2009) |
| [21] | Haus, H. A.; Islam, M. N., IEEE J. Quantum Electron., 21, 1172-1188 (1985) |
| [22] | Chen, Y.; Wu, M.; Tang, P.; Chen, S.; Du, J.; Jiang, G.; Li, Y.; Zhao, C.; Zhang, H.; Wen, S., Laser Phys. Lett., 11, 055101 (2014) |
| [23] | Strecker, K. E.; Partridge, G. B.; Truscott, A. G.; Hulet, R. G., Nature, 417, 150-153 (2002) |
| [24] | Weller, A.; Ronzheimer, J. P.; Gross, C.; Esteve, J.; Oberthaler, M. K., Phys. Rev. Lett., 101, 130401 (2008) |
| [25] | Theocharis, G.; Weller, A.; Ronzheimer, J. P.; Gross, C.; Oberthaler, M. K.; Kevrekidis, P. G.; Frantzeskakis, D. J., Phys. Rev. A, 81, 063604 (2010) |
| [26] | Ablowitz, M. J.; Kaup, D. J.; Newell, A. C.; Segur, H., Stud. Appl. Math., 53, 249-315 (1974) · Zbl 0408.35068 |
| [27] | Burtsev, S. P.; Zakharov, V. E.; Mikhailov, A. V., Theoret. Math. Phys., 70, 227-240 (1987) · Zbl 0639.35074 |
| [28] | Gordoa, P. R.; Pickering, A., J. Math. Phys., 40, 5749-5786 (1999) · Zbl 1063.37544 |
| [29] | Ning, T. K.; Chen, D. Y.; Zhang, D. J., Chaos Solitons Fractals, 21, 395-401 (2004) · Zbl 1049.35160 |
| [30] | Ning, T. K.; Chen, D. Y.; Zhang, D. J., Physica A, 339, 248-266 (2004) |
| [31] | Zhou, L. J., Phys. Lett. A, 34, 314-322 (2005) · Zbl 1345.35095 |
| [32] | Sun, Y. P.; Bi, J. B.; Chen, D. Y., Chaos Solitons Fractals, 26, 905-912 (2005) · Zbl 1093.35056 |
| [33] | Bi, J. B.; Sun, Y. P.; Chen, D. Y., Physica A, 364, 157-169 (2006) |
| [34] | Ning, T. K.; Zhang, W.; Chen, D. Y., Chaos Solitons Fractals, 34, 704-708 (2007) · Zbl 1134.37031 |
| [35] | Yao, Y. Q.; Chen, D. Y.; Zhang, D. J., Phys. Lett. A, 372, 2017-2025 (2008) · Zbl 1220.35145 |
| [36] | Zhou, L. J., Phys. Lett. A, 372, 5523-5528 (2008) · Zbl 1223.37091 |
| [37] | Ji, J., Nonlinear Anal., 71, 4034-4046 (2009) · Zbl 1173.35698 |
| [38] | Gu, C. H.; Hu, H. S.; Zhou, Z. X., Darboux Transformations in Integrable Systems—Theory and their Applications to Geometry (2005), Springer · Zbl 1084.37054 |
| [39] | Aktosun, T.; van der Mee, C., Inverse Problems, 25, 105003 (2009) · Zbl 1185.45006 |
| [40] | Aktosun, T.; van der Mee, C., AIP Conf. Proc., 1212, 254-263 (2010) |
| [41] | Cieslinski, J., J. Math. Phys., 36, 5670-5706 (1995) · Zbl 0845.35101 |
| [42] | Cieśliński, J. L., J. Phys. A: Math. Theor., 42, 404003 (2009) · Zbl 1193.37096 |
| [43] | Zhang, Y.; Deng, S. F.; Zhang, D. J.; Chen, D. Y., Physica A, 339, 228-236 (2004) |
| [44] | Zhao, D.; Zhang, Y. J.; Lou, W. W.; Luo, H. G., J. Math. Phys., 52, 043502 (2011) · Zbl 1316.35271 |
| [45] | He, J. S.; Zhang, L.; Chen, Y.; Li, Y. S., Sci. China Ser. A, 49, 1867-1878 (2006) · Zbl 1110.35302 |
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