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We study localized nonlinear excitations of a dilute Bose-Einstein condensate(BEC) with spin-orbit coupling in a deep optical lattice (OL). We use Wannierfunctions to derive a tight-binding model that includes the spin-orbit coupling(SOC) at the discrete level in the form of a generalized discrete nonlinearSch\"odinger equation. Spectral properties are investigated and the existenceand stability of discrete solitons and breathers with different symmetryproperties with respect to the OL is demonstrated. We show that the symmetry ofthe modes can be changed from on-site to inter-site and to asymmetric modessimply by changing the interspecies interaction. Asymmetric modes appear to benovel modes intrinsic of the SOC.

Discrete solitons of spin-orbit coupled Bose-Einstein condensates in optical lattices