Effects of constant and spatially varying higher-order dispersions on spatial solitons in \(\mathcal{P}\mathcal{T}\)-symmetric optical media under the alternative complex potentials

We investigated analytically and numerically the nonlinear Schrödinger (NLS) equation with constant and spatially varying third-order dispersion (TOD) in the alternative type of complex parity-time \((\mathcal{P}\mathcal{T})\) $(\mathcal{P}\mathcal{T})$ -symmetric potentials. This equation describes the propagation of ultra-short pulses through the optical media, where the real part of the potential models the index guiding and the imaginary part the loss/gain distribution of light within optical material. For the constant TOD, the regions of stability/instability linear \(\mathcal{P}\mathcal{T}\) $\mathcal{P}\mathcal{T}$ -symmetric phases are numerically carried out. By means of the linear stability analysis and direct numerical simulation, the effects of interplay between constant TOD and \(\mathcal{P}\mathcal{T}\) $\mathcal{P}\mathcal{T}$ -symmetric potential on the stability of these solutions are also tested. It is found that the constant TOD can be used to control the stability of these solutions. For the spatially varying TOD, the additive terms of the \(\mathcal{P}\mathcal{T}\) $\mathcal{P}\mathcal{T}$ -symmetric potential are considered for the nonlinear model and the robustness of these solutions against noise is tested by means of the split-step Fourier beam technic. Moreover, the elastic interactions of the two spatial solitons are generated under the \(\mathcal{P}\mathcal{T}\) $\mathcal{P}\mathcal{T}$ -symmetric potential for the spatially varying TOD. The power and the transverse power-flow density are further examined. Results indicate that the spatially varying TOD does not yield any instability in the self-focusing nonlinear medium with the chosen parameters values.