  In previous works, axion electrodynamics in three dimensional $p+is$
superconductors is discussed by borrowing the results from superconducting
Dirac systems. However, in this work, based on a systematic path integral
approach, we show that the axion electrodynamics in $p+is$ superconductors
exhibits a nonrelativistic form, which is different from the superconducting
Dirac and Weyl systems. More precisely, the induced electric field does not
enter into the axion action, and gauge invariance is ensured by the combination
of the electric potential and the Nambu-Goldstone phase mode. Furthermore,
unlike the axion angle in the Dirac case which is equal to the difference
between the superconducting phases on the two Fermi surfaces of different
helicities, the axion angle in the present case contains an additional
sinusoidal term. As applications of the derived nonrelativistic axion
electrodynamics, physical effects related to vortex lines and Witten effect are
discussed. Our work reveals the differences for axion electrodynamics between
the relativistic and nonrelativistic systems.
