  We study the shear momentum diffusion and related modes of a strongly coupled
$(2+1)$-dimensional conformal field theory at finite temperature and chemical
potential, using a dual holographic description. We consider a space-time
filling charged black brane solution of Einstein's gravity in
$(3+1)$-dimensional asymptotically Anti-de Sitter space coupled to a $U(1)$
gauge field via a Dirac-Born-Infeld action. In addition to temperature and
chemical potential, the holographic model has two other parameters: the tension
of the brane, and the non-linearity parameter controlling the higher-derivative
terms of the $U(1)$ field. By varying the parameters, one can, in particular,
interpolate between the Reissner-Nordstrom-AdS background and the background of
probe branes embedded into AdS space. We find analytically the retarded
two-point functions of the shear (transverse to the direction of spatial
momentum) components of the energy-momentum tensor and the global $U(1)$
current of the $(2+1)$-dimensional field theory in the hydrodynamic
approximation. We also find numerically the location of the poles of the
correlators (quasinormal modes) for a wide range of the parameters, focusing on
the effects of the back-reaction and non-linearities. We show, in particular,
that the shear diffusion constant agrees with the hydrodynamic form for a wide
range of parameters, including temperature and backreaction.
