Multivariable de Rham representations, Sen theory and $p$-adic differential equations

Let $K$ be a complete valued field extension of $\mathbf{Q}_p$ with perfect residue field. We consider $p$-adic representations of a finite product $G_{K,\Delta}=G_K^\Delta$ of the absolute Galois group $G_K$ of $K$. This product appears as the fundamental group of a product of diamonds. We develop the corresponding $p$-adic Hodge theory by constructing analogues of the classical period rings $\mathsf{B}_{\rm dR}$ and $\mathsf{B}_{\rm HT}$, and multivariable Sen theory. In particular, we associate to any $p$-adic representation $V$ of $G_{K,\Delta}$ an integrable $p$-adic differential system in several variables $\mathsf{D}_{\rm dif}(V)$. We prove that this system is trivial if and only if the representation $V$ is de Rham. Finally, we relate this differential system to the multivariable overconvergent $(\varphi,\Gamma)$-module of $V$ constructed by Pal and Z\'abr\'adi, along classical Berger's construction.