Bifurcation to multisoliton complexes in the ac-driven, damped nonlinear Schrödinger equation

We study bifurcations of localized stationary solutions of the externally driven, damped nonlinear Schr\"odinger equation $i{{Ψ}}ₜ+{{Ψ}}ₓₓ+2|{Ψ}{|}²{Ψ}={-}i{γ}{Ψ}{-}{he}^{i{Ω}t}$ in the region of large ${γ}$ $({γ}>1/2)$. For each pair of $h$ and ${γ}$, there are two coexisting solitons ${{Ψ}}₊$ and ${{Ψ}}_{{-}}.$ As the driver's strength $h$ increases for the fixed ${γ}$, the ${{Ψ}}₊$ soliton merges with the flat background while the ${{Ψ}}_{{-}}$ forms a stationary collective state with two ``${Ψ}$ pluses'': ${{Ψ}}_{{-}}{→}{{Ψ}}_{(+{-}+)}.$ We obtain other stationary solutions and identify them as multisoliton complexes ${{Ψ}}₍₊₊₎,{{Ψ}}_{({-}{-})},{{Ψ}}_{({-}+)},{{Ψ}}_{({-}{-}{-})},{{Ψ}}_{({-}+{-})},$ etc.