Geodesic
Darboux transformation and soliton solutions for the generalized coupled variable-coefficient nonlinear Schrödinger–Maxwell–Bloch system with symbolic computation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 4

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In an inhomogeneous nonlinear light guide doped with two-level resonant atoms, the generalized coupled variable-coefficient nonlinear Schrödinger–Maxwell–Bloch system can be used to describe the propagation of optical solitons. In this paper, the Lax pair and conservation laws of that model are derived via symbolic computation. Furthermore, based on the Lax pair obtained, the Darboux transformation is constructed and soliton solutions are presented. Figures are plotted to reveal the following dynamic features of the solitons: (1) Periodic mutual attractions and repulsions of four types of bound solitons: of two one-peak bright solitons; of two one-peak dark solitons; of two two-peak bright solitons and of two two-peak dark solitons; (2) Two types of elastic interactions of solitons: of two bright solitons and of two dark solitons; (3) Two types of parallel propagations of parabolic solitons: of two bright solitons and of two dark solitons. Those results might be useful in the study of optical solitons in some inhomogeneous nonlinear light guides. 
@article{ZVMMF_2012_52_4_a6,
     author = {Rui Guo and Bo Tian and Xing L\"u and Hai-Qiang Zhang and Wen-Jun Liu},
     title = {Darboux transformation and soliton solutions for the generalized coupled variable-coefficient nonlinear {Schr\"odinger{\textendash}Maxwell{\textendash}Bloch} system with symbolic computation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {671},
     publisher = {mathdoc},
     volume = {52},
     number = {4},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_4_a6/}
}
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Rui Guo; Bo Tian; Xing Lü; Hai-Qiang Zhang; Wen-Jun Liu. Darboux transformation and soliton solutions for the generalized coupled variable-coefficient nonlinear Schrödinger–Maxwell–Bloch system with symbolic computation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 4. http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_4_a6/