  This paper concerns with the asymptotic behavior of complete non-compact
convex curves embedded in $\mathbb{R}^2$ under the $\alpha$-curve shortening
flow for exponents $\alpha >\frac12$. We show that any such curve having in
addition its two ends asymptotic to two parallel lines, converges under
$\alpha$-curve shortening flow to the unique translating soliton whose ends are
asymptotic to the same parallel lines. This is a new result even in the
standard case $\alpha=1$, and we prove for all exponents up to the critical
case $\alpha>\frac12$.
