jats:titleAbstract</jats:title> jats:pWe investigate the Cauchy problem on the cylinder, namely the semi-periodic problem where there is periodicity in the jats:italicx</jats:italic>-direction and decay in the jats:italicy</jats:italic>-direction, for the Kadomtsev–Petviashvili II equation by the inverse spectral transform method. For initial data with small jats:italicL</jats:italic> jats:sup1</jats:sup> and jats:italicL</jats:italic> jats:sup2</jats:sup> 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.</jats:p>