M-shaped solitons in cubic nonlinear media with a composite linear potential

The M-shaped solitons—a special type of solitons—have been reported in many models, while most of them are not focused on the commonly used nonlinear Schrödinger equation (NLSE), which is widely employed to describe the propagation of laser beams in nonlinear media or the evolution of Bose–Einstein condensates. We demonstrate that families of M-shaped solitons can be generated in the NLSE with cubic (Kerr) nonlinear media and the help of a composite linear potential. It should be mentioned that these M-shaped solitons are completely stable for all values of the parameters of the model, a result obtained by the linear stability analysis and verified by the direct numerical simulations. The M-shaped solitons quickly disperse if the linear potential is taken away. We also report robust propagations of breather-like solitons in our model, when we use a specific soliton modulation procedure in the propagation of input M-shaped waveforms. The self-adaptive processes for M-shaped solitons are also investigated, and we show that these solitons will turn into irregular oscillations if the parameters of the external potential suffer a sudden variation. Interestingly, the solitons will keep their M-shaped waveforms and experience regular propagations if the parameters of the potential change gradually.