The inverse curve shortening flow on the hyperbolic plane

We study the inverse curve shortening flow in the hyperbolic plane ${\text{h}}^2$. We classify all solitons with respect to parabolic and conformal vector fields of ${\text{h}}^2$. In the upper half-plane model of ${\text{h}}^2$, we prove that parabolic solitons are all graphs on the $y$-axis, whereas conformal solitons are graphs on the $x$-axis. We study the concavity of these solitons and when they approach the coordinate axes.