Analytic study on certain solitons in an erbium-doped optical fibre

(2+1)-dimensional non-linear optical waves through the coherently excited resonant medium doped with the erbium atoms can be described by a (2+1)-dimensional non-linear Schrodinger equation coupled with the self-induced transparency equations. For such a system, via the Hirota method and symbolic computation, linear forms, one-, two- and N-soliton solutions are obtained. Asymptotic analysis is conducted and suggests that the interaction between the two solitons is elastic. Bright solitons are obtained for the fields E and P, while the dark ones for the field N, with E as the electric field, P as the polarization in the resonant medium induced by the electric field, and N as the population inversion profile of the dopant atoms. Head-on interaction between the bidirectional two solitons and overtaking interaction between the unidirectional two solitons are seen. Influence of the averaged natural frequency ω on the solitons are studied: (1) ω can affect the velocities of all the solitons; (2) Amplitu...