Amplitude analysis of $D^{+} \rightarrow K_{S}^{0} π^{+} π^{+} π^{-}$
The decay $D^{+} \rightarrow K_{S}^{0} π^{+} π^{+} π^{-}$ is studied with an amplitude analysis using a data set of 2.93${\mbox{\,fb}^{-1}}$ of $e^+e^+$ collisions at the $ψ(3770)$ peak accumulated by the BESIII detector. Intermediate states and non-resonant components, and their relative fractions and phases have been determined. The significant amplitudes, which contribute to the model that best fits the data, are composed of five quasi-two-body decays $ K_{S}^{0} a_{1}(1260)^{+}$, $ \bar{K}_{1}(1270)^{0} π^{+}$ $ \bar{K}_{1}(1400)^{0} π^{+}$, $ \bar{K}_{1}(1650)^{0} π^{+}$, and $ \bar{K}(1460)^{0} π^{+}$, a three-body decays $K_{S}^{0}π^{+}ρ^{0}$, as well as a non-resonant component $ K_{S}^{0}π^{+}π^{+}π^{-}$. The dominant amplitude is $ K_{S}^{0} a_{1}(1260)^{+}$, with a fit fraction of $(40.3\pm2.1\pm2.9)\%$, where the first and second uncertainties are statistical and systematic, respectively.