Semi-vortex solitons and their excited states in spin-orbit-coupled binary bosonic condensates

It is known that two-dimensional two-component fundamental solitons of the semi-vortex (SV) type, with vorticities $(s_{+},s_{-})=(0,1)$ in their components, are stable ground states (GSs) in the spin-orbit-coupled (SOC) binary Bose-Einstein condensate with the contact self-attraction acting in both components, in spite of the possibility of the critical collapse in the system. However, excited states(ESs) of the SV solitons, with the vorticity set $(s_{+},s_{-})=( S_{+},S_{+}+1)$ and $S_{+}=1,2,3,...$, are unstable in the same system. We construct ESs of SV solitons in the SOC system with opposite signs of the self-interaction in the two components. The main finding is stability of the ES-SV solitons, with the extra vorticity (at least) up to $S_{+}=6$. The threshold value of the norm for the onset of the critical collapse, $N_{\mathrm{thr}}$, in these excited states is higher than the commonly known critical value, $N_{c}\approx 5.85$,associated with the single-component Townes solitons, $N_{\mathrm{thr}}$ increasing with the growth of $S_{+}$. A velocity interval for stable motion of the GS-SV solitons is found too. The results suggest a solution for the challenging problem of the creation of stable vortex solitons with high topological charges.