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Giant magnon solution and dispersion relation in string theory in AdS_3 x S^3 x T^4 with mixed flux

We address the question about the exact form of the dispersion relation for light-cone string excitations in string theory in AdS3 x S3 x T4 with mixed R-R and NS-NS 3-form fluxes. The analogy with string theory in AdS5 x S5 suggests that in addition to the data provided by the perturbative near-BMN expansion and the symmetry algebra considerations there is also another source of information about the dispersion relation -- the semiclassical giant magnon solution. In earlier work in arXiv:1303.1037 and arXiv:1304.4099 it was found that the symmetry algebra constraints consistent with perturbative expansion do not completely determine the form of the dispersion relation. The aim of the present paper is to fix it by constructing a generalization of the known dyonic giant magnon soliton on S3 to the presence of a non-zero NS-NS flux described by a WZ term in the string action. We find that the angular momentum of this soliton gets shifted by a term linear in world-sheet momentum. We also discuss the symmetry algebra of the string light-cone S-matrix and show that the exact dispersion relation, which should have the correct perturbative BMN and semiclassical giant magnon limits, should also contain such a linear momentum term. The simplicity of the resulting bound-state picture provides a strong argument in favour of this dispersion relation.