Effective Lagrangians for BCS Superconductors at T=0
  We show that the low frequency, long wavelength dynamics of the phase of the pair field for a BCS-type s-wave superconductor at T=0 is equivalent to that of a time-dependent non-linear Schr\"odinger Lagrangian (TDNLSL), when terms required by Galilean invariance are included. If the modulus of the pair field is also allowed to vary, the system is equivalent to two coupled TDNLSL's.   We also refer the interested reader to our earlier paper, `Nonlinear Schrodinger equation for superconductors' [cond-mat/9312099], for a different line of derivation 
I.R. Aitchison, P. Ao, D.J. Thouless, and X.-M. Zhu