  By applying the recently proposed antisymmetric superfluid local density
approximation (ASLDA) to strongly interacting polarized atomic gases at
unitarity in very elongated traps, we find families of Larkin-Ovchinnikov (LO)
type of solutions with prominent transversal oscillation of pairing potential.
These LO states coexist with a superfluid state having a smooth pairing
potential. We suggest that the LO phase could be accessible experimentally by
increasing adiabatically the trap aspect ratio. We show that the local
asymmetry effects contained in ASLDA do not support a deformed superfluid core
predicted by previous Bogoliubov-de Gennes treatments.
