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 Schrödinger 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 $\mathit{\omega }$ on the solitons are studied: (1) $\mathit{\omega }$ can affect the velocities of all the solitons; (2) Amplitudes of the solitons for the fields P and N increase with $\mathit{\omega }$ decreasing, and decrease with $\mathit{\omega }$ increasing; (3) With $\mathit{\omega }$ decreasing, for the fields P and N, one-peak one soliton turns into the two-peak one, as well as interaction type changes from the interaction between two one-peak ones to that between a one-peak one and a two-peak one; (4) For the field E, influence of $\mathit{\omega }$ on the solitons cannot be found. The results of this paper might be of potential applications in the design of optical communication systems which can produce the bright and dark solitons simultaneously.