In this paper, we investigate D-term inflation within the framework of supergravity, employing the minimal Kähler potential. Following previous studies that revealed that this model can overcome the $\eta$-problem found in F-term models, we explore reheating dynamics and gravitino production, emphasizing the interplay between reheating temperature, spectral index, and gravitino abundance. Our analysis indicates that gravitino production is sensitive to the equation of state during reheating, affecting the reheating temperature and subsequent dark matter relic density. Furthermore, we analyze gravitational waves generated by cosmic strings, providing critical constraints on early Universe dynamics and cosmic string properties, the energy scales of both inflation and string formation influence the stochastic gravitational wave background (SGWB) generated by these cosmic strings.

We address effects of spin-orbit coupling (SOC), phenomenologically added to a two-component Bose-Einstein condensate composed of particles moving by Levy flights, in one- and two-dimensional (1D and 2D) settings. The corresponding system of coupled Gross-Pitaevskii equations includes fractional kinetic-energy operators, characterized by the Levy index, \alpha < 2 (the normal kinetic energy corresponds to \alpha = 2). The SOC terms, with strength \lambda, produce strong effects in the 2D case: they create families of stable solitons of the semi-vortex (SV) and mixed-mode (MM) types in the interval of 1 < \alpha < 2, where the supercritical collapse does not admit the existence of stable solitons in the absence of the SOC. At \lambda --> 0, amplitudes of these solitons vanish as (\lambda)^{1/(\alpha - 1)}.