  A pair density wave (PDW) is a superconductor whose order parameter is a
periodic function of space, without an accompanying spatially-uniform
component. Since PDWs are not the outcome of a weak-coupling instability of a
Fermi liquid, a generic pairing mechanism for PDW order has remained elusive.
We describe and solve models having robust PDW phases. To access the
intermediate coupling limit, we invoke large $N$ limits of Fermi liquids with
repulsive BCS interactions that admit saddle point solutions. We show that the
requirements for long range PDW order are that the repulsive BCS couplings must
be non-monotonic in space and that their strength must exceed a threshold
value. We obtain a phase diagram with both finite temperature transitions to
PDW order, and a $T=0$ quantum critical point, where non-Fermi liquid behavior
occurs.
