Analytic solutions for the (2+1)-dimensional generalized sine-Gordon equations in nonlinear optics

The (2+1)-dimensional generalized sine-Gordon equations can be used to describe the propagation of femtosecond laser pulse in a systems of two-level atoms. In this paper, bilinear forms of the equations have been constructed through variable transformation and Bell polynomial. Based on the bilinear method and truncated Painlevé expansion, multi-soliton and quasi-periodic peakon solutions for such a system are derived, respectively. Overtaking elastic interactions between the two solitons and among the three solitons have been found. It is presented that the soliton with the smaller amplitude can travel faster than the larger one. Apart from the soliton, other type of nonlinear wave propagating in the systems of two-level atoms has been displayed, namely the quasi-periodic peakon. The peakon oscillates in both space and time, while its left and right derivatives at peak point are relatively large, which means the electric-field intensity of the femtosecond laser pulse is rather strong.