  We find explicit solutions for giant magnons and spiky strings living on the
Schr\"odinger $Schr_5 \times T^{1,1}$ and compute dispersion relations. The
holographic dual field theory is conjectured to be a non-local dipole-deformed
CFT at strong coupling. We find that the dependence between conserved charges
in the dispersion relations is transcendental, which is quite different from
the most symmetric case of spherical internal space. Keeping the squashing
parameter $b$ general allows us to take some limits and to compare our results
to known cases.
