  We consider the focusing $L^2$-critical half-wave equation in one space
dimension $$ i \partial_t u = D u - |u|^2 u, $$ where $D$ denotes the
first-order fractional derivative. Standard arguments show that there is a
critical threshold $M_* > 0$ such that all $H^{1/2}$ solutions with $\| u
\|_{L^2} < M_*$ extend globally in time, while solutions with $\| u \|_{L^2}
\geq M_*$ may develop singularities in finite time.
  In this paper, we first prove the existence of a family of traveling waves
with subcritical arbitrarily small mass. We then give a second example of
nondispersive dynamics and show the existence of finite-time blowup solutions
with minimal mass $\| u_0 \|_{L^2} = M_*$. More precisely, we construct a
family of minimal mass blowup solutions that are parametrized by the energy
$E_0 >0$ and the linear momentum $P_0 \in \R$. In particular, our main result
(and its proof) can be seen as a model scenario of minimal mass blowup for
$L^2$-critical nonlinear PDE with nonlocal dispersion.
