[Paper Review] Discrete curves in CP1 and the Toda lattice

Tim Hoffmann, Nadja Kutz|ArXiv.org|Aug 23, 2002

Advanced Algebra and Geometry8 references5 citations

TL;DR

This paper introduces a geometric interpretation of the one-dimensional Toda lattice hierarchy and its reductions as flows on discrete curves in ℂP¹, using homogeneous coordinates in ℂ². By relating Flaschka-Manakov variables to cross-ratios and curvature invariants, it shows that quadric curves are invariant under trivial Toda flows and discrete planar elastic curves are invariant under the discrete mKdV flow, providing a novel geometric realization of integrable systems via discrete differential geometry.

ABSTRACT

In this paper we investigate flows on discrete curves in $\C^2$, $\CP^1$, and $\C$. A novel interpretation of the one dimensional Toda lattice hierarchy and reductions thereof as flows on discrete curves will be given.

Motivation & Objective

• To establish a geometric framework for the Toda lattice hierarchy using discrete curves in ℂP¹.
• To interpret the Toda lattice and its reductions (e.g., Volterra hierarchy) as flows on discrete curves with specific parametrization.
• To identify classes of discrete curves invariant under Toda and mKdV flows, particularly under Euclidean and tangential transformations.
• To connect discrete curvature and cross-ratios to integrable system equations, such as the discrete KdV and mKdV equations.
• To explore the potential for extending this geometric approach to discrete surfaces and generalized Toda systems.

Proposed method

• Lift discrete curves in ℂP¹ to homogeneous coordinates in ℂ², using maps γ: ℤ → ℂ² with γₖ = (xₖ, yₖ) and cₖ = xₖ/yₖ.
• Define key geometric invariants: gₖ = det(γₖ, γₖ₊₁) for conformal arc length deviation and uₖ = det(γₖ₋₁, γₖ₊₁) for cross-ratio structure.
• Use recursive reconstruction via Lemma 1: γₖ₊₁ = (1/gₖ₋₁)(uₖγₖ - gₖγₖ₋₁), showing that uₖ and gₖ fully determine the curve.
• Introduce a zero curvature (Lax) representation for flows on curves, linking evolution equations to the Toda lattice hierarchy.
• Apply Euclidean reduction by assuming yₖ ≠ 0, projecting curves from ℂ² to ℂ, and deriving the discrete mKdV flow from the second Volterra flow.
• Use curvature evolution equations to define discrete planar elastic curves as critical points of a functional involving log(1 + κₖ/4), leading to a discrete mKdV-compatible equation.

Experimental results

Research questions

• RQ1 How can the one-dimensional Toda lattice hierarchy be geometrically interpreted as flows on discrete curves in ℂP¹?
• RQ2 What is the role of cross-ratios and Flaschka-Manakov variables in characterizing discrete curve flows?
• RQ3 Which discrete curves remain invariant under the trivial Toda flow (pₖ = 0) and the discrete mKdV flow?
• RQ4 How does the Euclidean reduction of arc length parametrized curves in ℂ² yield a discrete mKdV flow on curves in ℂ?
• RQ5 What is the geometric significance of discrete elastic curves in the context of integrable curve flows?

Key findings

• Quadric curves in ℂ² are invariant under the trivial Toda flow (pₖ = 0), corresponding to constant Flaschka-Manakov variables and zero curvature evolution.
• The cross-ratios of arc length parametrized discrete curves evolve according to the Volterra hierarchy, with the first flow corresponding to the discrete KdV equation.
• Under Euclidean reduction, curves satisfying the discrete mKdV flow evolve by translation and tangential flow, and discrete planar elastic curves are invariant under this flow.
• The curvature of discrete elastic curves satisfies the equation κₖ₊₁ = 2aκₖ/(1 + κₖ²/4) - κₖ₋₁, which matches the discrete mKdV equation for a = 1.
• A generalized curvature evolution equation κₖ₊₁ = 2aκₖ/(1 + κₖ²/4) - κₖ₋₁ + b + cₖ with cₖ₊₁ = -cₖ describes curves invariant under mKdV flow up to tangential and Euclidean transformations.
• The discrete mKdV flow arises geometrically from the second flow of the Volterra hierarchy when restricted to Euclidean-reduced curves in ℂ, providing a discrete analog of the Miura transformation.