  We investigate the Cauchy problem on the cylinder, namely the semi-periodic
problem where there is periodicity in the $x$-direction and decay in the
$y$-direction, for the Kadomtsev-Petviashvili II equation by the inverse
spectral transform method. For initial data with small $L^1$ and $L^2$ norms,
assuming the zero mass constraint, this initial-value problem is reduced to a
Riemann-Hilbert problem on the boundary of certain infinite strips with shift.
