  Recently, Bi-based compounds have attracted attentions because of the strong
spin-orbit coupling (SOC). In this work, we figured out the role of SOC in
ABi$_{3}$ (A=Sr and Ba) by theoretical investigation of the band structures,
phonon properties, and electron-phonon coupling. Without SOC, strong Fermi
surface nesting leads to phonon instabilities in ABi$_{3}$. SOC suppresses the
nesting and stabilizes the structure. Moreover, without SOC the calculation
largely underestimates the superconducting transition temperatures ($T_{c}$),
while with SOC the calculated $T_{c}$ are very close to those determined by
measurements on single crystal samples. The SOC enhanced superconductivity in
ABi$_{3}$ is due to not only the SOC induced phonon softening, but also the SOC
related increase of electron-phonon coupling matrix elements. ABi$_{3}$ can be
potential platforms to construct heterostructure of superconductor/topological
insulator to realize topological superconductivity.
