  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.
