Improved amplitude analysis of ${\eta }^{\mathrm{\prime }}\to {\pi }^{+}{\pi }^{-}{\pi }^0$ and ${\eta }^{\mathrm{\prime }}\to {\pi }^0{\pi }^0{\pi }^0$

Using a sample of $(10087\pm 44)\times {10}^6$ $J\mathrm{/}\psi$ events collected with the BESIII detector at BEPCII, we perform an amplitude analysis of the decays ${\eta }^{\mathrm{\prime }}\to {\pi }^{+}{\pi }^{-}{\pi }^0$ and ${\eta }^{\mathrm{\prime }}\to {\pi }^0{\pi }^0{\pi }^0$, where we observe significant ${\pi }^{\pm }{\pi }^0$ $P$-wave and $\pi$-$\pi$ $S$-wave interactions. Two different parameterizations, a $\pi$-$\pi$ scattering phase shift and the Gounaris-Sakurai Breit-Wigner formalism, are used to describe the $P$-wave propagator. Due to the large interference, the branching fractions for both the $P$- and the $S$-waves are found to be strongly model dependent.