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Darboux's Theorem in $p$-adic symplectic geometry

Luis Crespo, Álvaro Pelayo

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 Schrödinger equation. Symplectic geometry is deeply connected with classical and quantum mechanics, and a main motivation of this paper is to pursue a $p$-adic version of symplectic geometry following recent developments which incorporate the $p$-adic numbers into mathematical and theoretical physics.

Darboux's Theorem in $p$-adic symplectic geometry

Abstract

Let be a prime number. We derive an analog of Moser's Path Method for -adic analytic manifolds and use it to prove a -adic analog of Darboux's Theorem. Using it as a stepping stone we give a classification of second-countable -adic analytic symplectic manifolds in terms of -adic volume. This is a symplectic version of a classical result of Serre in -adic analytic geometry from 1965. We also prove a -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 Schrödinger equation. Symplectic geometry is deeply connected with classical and quantum mechanics, and a main motivation of this paper is to pursue a -adic version of symplectic geometry following recent developments which incorporate the -adic numbers into mathematical and theoretical physics.

Paper Structure

This paper contains 18 sections, 17 theorems, 113 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. $p$-adic Moser's Path Method
  3. Applications of $p$-adic Moser's Path Method to symplectic geometry
  4. Applications to local symplectic geometry
  5. Applications to global symplectic geometry
  6. Application of the $p$-adic Moser's Path Method to the Ablowitz-Ladik and Salerno models
  7. Technical results on $p$-adic initial value problems, power series and forms on manifolds
  8. Generalization of Moser's Path Method to $p$-adic geometry: proof of Theorem \ref{['thm:moser']}
  9. The Darboux theorem on $p$-adic analytic symplectic manifolds: proof of Theorem \ref{['thm:darboux2']}
  10. The Darboux-Weinstein theorem on $p$-adic analytic symplectic manifolds: proof of Theorem \ref{['thm:darboux-weinstein']}
  11. Global classification of $p$-adic analytic symplectic manifolds: proof of Theorems \ref{['thm:volumes']} and \ref{['thm:global-coord']}
  12. Application to nonlinear Schrödinger equation
  13. Final remarks and context of the paper
  14. $p$-adic numbers and $p$-adic geometry
  15. $p$-adic numbers, power series and analytic functions
  16. ...and 3 more sections

Key Result

Theorem A

Let $\ell$ and $k$ be integers with $0\leqslant k\leqslant \ell$ and $\ell\geqslant 1$. Let $p$ be a prime number. Let $\mathbb{Z}_p$ denote the $p$-adic integers. Let $d=2$ if $p=2$ and otherwise $d=1$. Let $M$ be an $\ell$-dimensional $p$-adic analytic manifold and let $Q$ be a compact submanifold Let $\mathrm{L}$ denote the $p$-adic Lie derivative and suppose that the differential equation has

Figures (8)

  • Figure 1: Illustration of the $p$-adic analytic Darboux's Theorem (Theorem \ref{['thm:darboux2']}).
  • Figure 2: $p$-adic tubular neighborhoods of dimension $0$, $1$ and $2$ submanifolds of a dimension $3$ manifold as in Definition \ref{['def:tubular']}.
  • Figure 3: Illustration of the proof of Theorem \ref{['thm:darboux']}. First figure: the form $\alpha=\omega_1-\omega_0$ is exact on $U$, that is, it is $\mathrm{d}\beta$ for some $1$-form $\beta$, and this form is given by a power series in a subset $U_2\subset U$. Second figure: we determine $S$ as a subset of $U_2\times T$ and a subset $U_3\subset U_2$ that avoids the points in $S$. Third figure: the vector field $X_t$ on $U_3$. Fourth figure: the resulting sets $U_0$ and $U_1$.
  • Figure 4: Steps of the proof of Theorem \ref{['thm:darboux-weinstein']}. First figure: the manifold $M$, the compact submanifold $Q$, and a tubular neighborhood $U$ of $Q$. Second figure: open subsets $U_1,U_2,U_3,U_4$ covering $Q$ such that $\omega_0$ and $\beta$ are given as power series. Third figure: we reduce $U_i$ to $U_i'$ to avoid the points in $S_i$. Fourth figure: the vector field $X_{i,t}$ in $U_i'$ and the resulting set $U_i"$. Fifth figure: the set $U_0$ is the union of all $U_i"$.
  • Figure 5: The balls $\mathrm{B}_{ki}$, for $p=3$, $d=2$, $0\leqslant k\leqslant 2$, and $1\leqslant i\leqslant 8$. The ball $\mathrm{B}_{ki}$ has radius $p^k$ and a color which depends on $i$.
  • ...and 3 more figures

Theorems & Definitions (36)

  • Theorem A: $p$-adic analytic Moser's Path Method
  • Theorem B: $p$-adic analytic Darboux's Theorem
  • Theorem C: $p$-adic analytic Darboux-Weinstein's Theorem
  • Theorem D
  • Theorem E: Symplectic Serre's classification of $p$-adic analytic manifolds
  • Theorem F
  • Lemma 2.1: Multivariable $p$-adic initial value problem
  • proof
  • Remark 2.2
  • Example 2.3
  • ...and 26 more