Zeros of OPUC and Long Time Asymptotics of Schur and Related Flows

We provide a complete analysis of the asymptotics for the semi-infinite Schur flow: $\alpha_j(t)=(1- |\alpha_j(t)|^2) (\alpha_{j+1}(t)-\alpha_{j-1}(t))$ for $\alpha_{-1}(t)= 1$ boundary conditions and $n=0,1,2,...$, with initial condition $\alpha_j(0)\in (-1,1)$. We also provide examples with $\alpha_j(0)\in\mathbb{D}$ for which $\alpha_0(t)$ does not have a limit. The proofs depend on the solution via a direct/inverse spectral transform.