Gaussian soliton and periodic wave solutions with their stabilities in the cubic-quintic Gross–Pitaevskii equation integrating spin-orbit momentum effects

This paper investigates the presence and stability of nonlinear localized modes within the Gross–Pitaevskii Equation (GPE), considering interactions involving cubic–quintic nonlinearities and varying spin-orbit momentum (SOM). It also explores two distinct types of complex parity-time ( $$\mathcal{P}\mathcal{T}$$ P T )-symmetric potentials, specifically Gaussian harmonic and periodic potentials. The influence of the SOM coefficient on regions of unbroken and broken phases is examined, revealing its modulation effect on the nonlinear stability and power distribution of these modes. Additionally, the interaction dynamics of two spatial solitons are analyzed within the context of the $$\mathcal{P}\mathcal{T}$$ P T -symmetric Gaussian potential. Notably, it is found that solitons remain stable even when the $$\mathcal{P}\mathcal{T}$$ P T -symmetry of the underlying nonlinear model is disrupted. The accuracy of the findings is confirmed through comparisons with numerical simulations and exact analytical expressions of the localized modes in one dimension (1D). The numerical simulations also indicate that obtaining the stable solitons of the cubic–quintic GPE with a varying SOM term is most challenging when the considered $$\mathcal{P}\mathcal{T}$$ P T -symmetric potential is periodic. Graphical abstract