Finite-size giant magnons on \eta-deformed AdS_5 x S^5
  We consider strings moving in the R_t x S^3_\eta subspace of the \eta-deformed AdS_5 x S^5 and obtain a class of solutions depending on several parameters. They are characterized by the string energy and two angular momenta. Finite-size dyonic giant magnon belongs to this class of solutions. Further on, we restrict ourselves to the case of giant magnon with one nonzero angular momentum, and obtain the leading finite-size correction to the dispersion relation. 
Changrim Ahn and Plamen Bozhilov
hep-th