  We prove global well-posedness of the short-pulse equation with small initial
data in Sobolev space $H^2$. Our analysis relies on local well-posedness
results of Sch\"afer & Wayne, the correspondence of the short-pulse equation to
the sine-Gordon equation in characteristic coordinates, and a number of
conserved quantities of the short-pulse equation. We also prove local and
global well-posedness of the sine-Gordon equation in an appropriate function
space.
