  We produce the open strings on $\mathbb{R}\times S^{2}$ that correspond to
the solutions of integrable boundary sine-Gordon theory by making use of the
$N$-magnon solutions provided in \cite{KPV} together with explicit moduli.
Relating the two boundary parameters in a special way we describe the
scattering of giant magnons with non-maximal $Y=0$ giant gravitons and
calculate the leading contribution to the associated magnon scattering phase.
