📝 SummarXiv

Abstract

Let $p$ be a prime number. We derive an analog of Moser's Path Method for $p$-adic analytic manifolds and use it to prove a $p$-adic analog of Darboux's Theorem. Using it as a stepping stone we give a classification of second-countable $p$-adic analytic symplectic manifolds in terms of $p$-adic volume. This is a symplectic version of a classical result of Serre in $p$-adic analytic geometry from 1965. We also prove a $p$-adic version of Weinstein's generalization of Darboux's Theorem to neighborhoods of compact manifolds. Finally we find explicitly Darboux's coordinates for the physical models by Ablowitz-Ladik and Salerno of the Discrete Nonlinear Schr\"odinger equation.