The formula for the self ionization of water is 2H₂O(l)⇄H₃O⁺(aq)+OH⁻(aq)
The hydronium (H₃O⁺) is usually just referred to as a hydrogen ion or a proton (H⁺) and hydroxide (OH⁻) doesn't have another name that I am aware of. These ions do stay in solution. However the concentrations are really small and the equilibrium constant (K(w)) is 1×10⁻¹⁴.
I hope this helps. Let me know if anything is unclear.