Non-autonomous solitons in inhomogeneous nonlinear media with distributed dispersion

We show that the methodology based on the powerful method of the so-called hidden symmetry reductions (HSR) provides a systematic way to analytically derive non-autonomous temporal soliton solutions of a one-dimensional inhomogeneous nonlinear Schrodinger equation with a linear gain or loss, a linear density profile, and distributed dispersion. The fundamental innovation of our approach is to notice that the ansatz used in the powerful method of the HSR contains a free parameter that can be used to control the wave propagations related to both the rogue and the symmetric cnoidal solitons raising the possibility of relative experiments and potential applications in nonlinear optics and Bose-Einstein condensates (BECs). We show that for BECs with the time-dependent s-wave scattering length in linear potential when the loss or the gain of atoms is taken into consideration our algorithm is a useful tool to control non-autonomous bright rogue and symmetric cnoidal solitons of BECs. Our results show that the nonlinearity and the linear trapping potential play important roles in the evolutional characteristics such as amplitude, width, trajectory, linear chirp phase, phase offset, and phase shift. Our analysis might have the applications in the theory of nonlinear transmission networks for achieving pulse compression.