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Periodic discrete Darboux transforms

Joseph Cho, Katrin Leschke, Yuta Ogata

TL;DR

The paper addresses global questions in discrete differential geometry by studying periodic Darboux transforms of discrete polarised curves. It develops a quaternionic, gauge-theoretic framework in which Darboux transforms correspond to parallel sections of a one-parameter family of flat connections, enabling linearisation of monodromy and a coherent discrete analogue of the smooth theory. It establishes discrete Darboux transforms via cross-ratio conditions, connects them to discrete Riccati equations, and proves monodromy results including discrete bicycle correspondences, with explicit closed-form parametrisations for discrete circles and circletons. The approach yields efficient recurrence-based constructions for periodic solutions of integrable systems and has potential applications to modeling periodic phenomena in physics and geometry.

Abstract

We express Darboux transformations of discrete polarised curves as parallel sections of discrete connections in the quaternionic formalism. This immediately leads to the linearisation of the monodromy of the transformation. We also consider the integrable reduction to the case of discrete bicycle correspondence. Applying our method to the case of discrete circles, we obtain closed-form discrete parametrisations of all (closed) Darboux transforms and (closed) bicycle correspondences.

Periodic discrete Darboux transforms

TL;DR

The paper addresses global questions in discrete differential geometry by studying periodic Darboux transforms of discrete polarised curves. It develops a quaternionic, gauge-theoretic framework in which Darboux transforms correspond to parallel sections of a one-parameter family of flat connections, enabling linearisation of monodromy and a coherent discrete analogue of the smooth theory. It establishes discrete Darboux transforms via cross-ratio conditions, connects them to discrete Riccati equations, and proves monodromy results including discrete bicycle correspondences, with explicit closed-form parametrisations for discrete circles and circletons. The approach yields efficient recurrence-based constructions for periodic solutions of integrable systems and has potential applications to modeling periodic phenomena in physics and geometry.

Abstract

We express Darboux transformations of discrete polarised curves as parallel sections of discrete connections in the quaternionic formalism. This immediately leads to the linearisation of the monodromy of the transformation. We also consider the integrable reduction to the case of discrete bicycle correspondence. Applying our method to the case of discrete circles, we obtain closed-form discrete parametrisations of all (closed) Darboux transforms and (closed) bicycle correspondences.
Paper Structure (11 sections, 11 theorems, 111 equations, 16 figures)

This paper contains 11 sections, 11 theorems, 111 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Darboux transformations of smooth space curves
  3. Darboux transformations via parallel sections
  4. Monodromy of Darboux transforms
  5. Arc-length polarisations and the bicycle monodromy
  6. Monodromy of discrete Darboux transformations
  7. Flat connection of a discrete polarised curve
  8. Discrete Darboux transformations via parallel sections
  9. Discrete monodromy
  10. Discrete bicycle correspondence and bicycle monodromy
  11. Summary

Key Result

Lemma 2.4

Given a polarised curve $x : (I, q) \to \mathbb{H}$, we have $\hat{x} := x + \alpha \beta^{-1}$ is a Darboux transform of $x$ with parameter $\mu$ if and only if $\phi := \left(\right)$ is $\mathcal{D}_\mu$--parallel.

Figures (16)

  • Figure 1: Closed Darboux transformations (or bicycle correspondences) of a discrete circle.
  • Figure 2: Original $(2,3)$-torus knot (in black), and its closed Darboux transforms (on the left), also viewed from the top (on the right).
  • Figure 3: Original planar curve given by $x(t) = \mathbbm{j} \cos(3t) e^{i t}$ (in black), and its closed Darboux transforms.
  • Figure 4: Darboux transforms of the circle (drawn in black) at the resonance point with $k = 3$. On the left are those in the $3$--space with $c_0^+ = 0.5\mathbbm{i}, c_0^- = 0, c_1^+ = 1, c_1^- = -4, -2, -1.2, -0.1, 0.25$, on the right are those in the plane with the same constants except $c_0^+ = 0$.
  • Figure 5: Trivial case of Darboux transformation keeping arc-length polarisation in $3$--space.
  • ...and 11 more figures

Theorems & Definitions (39)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4: cf. hertrich-jeromin_introduction_2003
  • proof
  • Theorem 2.5: cf. hertrich-jeromin_introduction_2003
  • Remark 2.6
  • Lemma 2.7: cf. hertrich-jeromin_introduction_2003
  • proof
  • Corollary 2.8
  • ...and 29 more