  Motivated by recent experiments in low-dimensional trapped fermionic
superfluids, we study a quasi-one-dimensional (quasi-1D) superfluid with a
population imbalance between two hyperfine states using an exact mean-field
solution for the order parameter. When an effective "magnetic field" exceeds a
critical value, the superfluid order parameter develops spatial inhomogeneity
in the form of a soliton lattice. The soliton lattice generates a band of
quasiparticle states inside the energy gap, which originate from the Andreev
bound states localized at the solitons. Emergence of the soliton lattice is
accompanied by formation of a spin-density wave, with the majority fermions
residing at the points in space where the Fulde-Ferrell-Larkin-Ovchinnikov
(FFLO) order parameter vanishes. We discuss possibilities for experimental
detection of the quasi-1D FFLO state using elastic and inelastic optical Bragg
scattering and radio-frequency spectroscopy. We show that these measurements
can provide necessary information for unambiguous identification of the
spatially-inhomogeneous quasi-1D FFLO state and the soliton lattice formation.
