In general, any time you consistently see doublets of your recombinant protein of interest, it's either truncated/proteolyzed protein or a covalently modified form. Cutting out the bands and sending them for MS sequencing is likely to be your most informative route.
Are you boiling your SDS-PAGE samples with mercaptoethanol prior to running on the gel? If you're not boiling, the doublet could be denatured and partially denatured protein. I've seen situations where 2% SDS does not completely denature the protein at room temp. If you're not using BME, you could have an internal disulfide.
If the lower band is truncated protein caused by premature translational terminatio, use a c-terminal affinity tag.
If the truncated protein caused by proteolysis, use a protease inhibitor cocktail during purification and you should get less of the lower band
If none of the prove useful, you likely have some kind of PTM (internal crosslink, lipidation, oxidation, etc).
Any chance you're expressing your protein with some kind of N-terminal signal sequence on it? We commonly express proteins that, in the native host, reside in the periplasm and are directed that way via an n-terminal 50 amino acid periplasmic signaling sequence that gets cleaved as it gets transported into the periplasm. When overexpressing these proteins in ecoli, we remove these first 50 or so amino acids from the expression construct otherwise we get strange expression results.