  Weyl fermions are massless chiral quasiparticles existing in materials known
as Weyl semimetals. Topological surface states, associated with the unusual
electronic structure in the Weyl semimetals, have been recently demonstrated in
linear systems. Ultracold atomic gases, featuring laser-assisted tunneling in
three-dimensional optical lattices, can be used for the emulation of Weyl
semimetals, including nonlinear effects induced by the collisional nonlinearity
of atomic Bose-Einstein condensates. We demonstrate that this setting gives
rise to topological states in the form of Weyl solitons at the surface of the
underlying optical lattice. These nonlinear modes, being exceptionally robust,
bifurcate from linear states for a given quasi-momentum. The Weyl solitons may
be used to design an efficient control scheme for topologically-protected
unidirectional propagation of excitations in light-matter-interaction physics.
After the recently introduced Majorana and Dirac solitons, the Weyl solitons
proposed in this work constitute the third (and the last) member in this family
of topological solitons.
