Open giant magnons suspended between dual giant gravitons in $\mathcal{N}=4$ SYM

We study classical solutions to the Nambu-Goto string on $Ad{S}_2\times S^1$ and $Ad{S}_3\times S^1$ corresponding to strings stretched between wrapped branes extending in $AdS$. The solutions are obtained by analytic continuation of giant magnon solutions, cut at the position of the branes. These solutions carry one or two $SO(2,4)$ charges and a single $SO(6)$ charge. We compute their energies and show their relation to $\frac{1}{2}$-BPS geometries. Their relevance to the $SL(2)$ sector of $\mathcal{N}=4$ SYM is also discussed.