The isometric rigidity of Wasserstein spaces and PDEs are inherently interconnected through the optimal transport theory, with the Monge-Amp\`{e}re equation serving as the core link. In this talk, based on the work of Geher, Titkos, and Virosztek, we extend their rigidity results on $[0,1]$\to $[0,1]$ to different bounded intervals. Paticularly, we give the characterization of isometries of $p = 1$ and $p > 1$, and we find the exotic isometries for $p=1$.