  Topological lasers based on topologically protected edge states offer unique
features and enhanced robustness of operation in comparison with conventional
lasers, even in the presence of disorder, edge deformation, and localized
defects. Here we propose a new class of topological lasers arising from the
valley Hall edge states, which does not require external magnetic fields or
dynamical modulations of the device parameters. Specifically, topological
lasing occurs at domain walls between two honeycomb waveguide arrays with
broken spatial inversion symmetry. Two types of valley Hall edge lasing modes
are found by shaping the gain landscape along the domain walls. In the presence
of uniform losses and two-photon absorption, lasing in edge states results in
the formation of stable nonlinear dissipative excitations localized on the edge
of the structure, even if it has complex geometry and even if it is finite.
Robustness of lasing states is demonstrated in both periodic and finite
structures, where such states can circumvent sharp corners without scattering
loss or radiation into the bulk. The photonic structure and mechanism proposed
here for topological lasing is fundamentally different from those previously
demonstrated topological lasers and can be used for fabrication of practical
topological lasers of various geometries.
