[@buzdin_organic_1984; @buzdin_nonuniform_1987; @croitoru_peculiarities_2014]. Such compounds consist of weakly coupled 1D chains. The interchain coupling is described by a hopping parameter $t$. If the hopping integral $t \ll T_c$, then the system can be described by a strictly 1D model. On the other hand, as it has been demonstrated in [@tsuzuki_long-range_1972], the mean-field treatment is justified if $t \gg T_c^2/E_\text{F}$. So for $T_c^2/E_\text{F} \ll t \ll T_c$, the critical fluctuations of the superconducting order parameter are effectively suppressed, and the system can be treated as a strictly superconducting 1D wire.

The paper is organized as follows. In section \[sec:weak\_soc\], we derive the Gor’kov equations [@gorkov_energy_1958; @abrikosov_methods_1965] for our system, and solve them to obtain the anomalous Gor’kov Green’s function Eq. (\[eq. Fdu-single\]) describing superconducting correlations for arbitrary exchange field and spin-orbit coupling. This solution is used to write the self-consistency gap equation in the regime of small spin-orbit fields $\alpha\,p_\text{F} \ll T_c$, and demonstrate the presence of a modulated superconducting phase. The modulation wavevector scales as $h_x^2\,h_z\,\alpha^3$. In section \[sec:strong\_soc\], we consider the regime of large spin-orbit coupling $\alpha\,p_\text{F} \gg T_c$. In the strong spin-orbit coupling regime, the electronic spectrum is drastically modified, and it is necessary to work in the 