  Chiral magnets possess topological line excitations where the magnetization
within each cross section forms a skyrmion texture. We study analytically and
numerically the low-energy, non-linear dynamics of such a skyrmion string in a
field-polarized cubic chiral magnet, and we demonstrate that it supports
solitary waves. Theses waves are in general non-reciprocal, i.e., their
properties depend on the sign of their velocity $v$, but this non-reciprocity
diminishes with decreasing $|v|$. An effective field-theoretical description of
the solitary waves is derived that is valid in the limit $v \to 0$ and gives
access to their profiles and their existence regime. Our analytical results are
quantitatively confirmed with micromagnetic simulations for parameters
appropriate for the chiral magnet FeGe. Similarities with solitary waves found
in vortex filaments of fluids are pointed out.
