[Paper Review] Tropical Jacobian and the generic fiber of the ultra-discrete periodic Toda lattice are isomorphic

Rei Inoue, Tomoyuki Takenawa|ArXiv.org|Feb 3, 2009

Polynomial and algebraic computation13 references3 citations

TL;DR

This paper proves that the generic isolevel set of the ultra-discrete periodic Toda lattice is isomorphic to the tropical Jacobian of its tropical spectral curve, establishing a complete solution via tropical theta functions. The isomorphism is constructed via a continuous, bijective map between the phase space and the tropical Jacobian, enabling a systematic solution to the initial value problem using quasi-periodic functions and lattice point analysis in the tropical setting.

ABSTRACT

We prove that the general isolevel set of the ultra-discrete periodic Toda lattice is isomorphic to the tropical Jacobian associated with the tropical spectral curve. This result implies that the theta function solution obtained in the authors' previous paper is the complete solution. We also propose a method to solve the initial value problem.

Motivation & Objective

• To establish a rigorous isomorphism between the generic isolevel set of the ultra-discrete periodic Toda lattice and the tropical Jacobian of its tropical spectral curve.
• To confirm that the theta function solution previously proposed is complete, by showing the isomorphism covers all solutions in the generic case.
• To provide a constructive method for solving the initial value problem of the ultra-discrete periodic Toda lattice using tropical geometric techniques.
• To extend the correspondence between integrable systems and tropical geometry by linking the dynamics of the UD-pToda to the algebraic structure of the tropical Jacobian.

Proposed method

• Construct a continuous, injective map ισ from the tropical Jacobian J(K) to the isolevel set TC of the ultra-discrete periodic Toda lattice.
• Use a quasi-periodic function Tnt satisfying a Hirota-type bilinear equation: Tnt−1 + Tnt+1 = min[2Tnt, Tn−1t+1 + Tn+1t−1 + L].
• Define the phase space variables Qnt and Wnt via a transformation involving Tnt and Tnt+1, with a shift parameter d and a linear combination of Tnt terms.
• Apply a limiting procedure with a large parameter δ to isolate the dominant terms in the tropical theta function, enabling identification of the region Dm in which the point Znt lies.
• Use the asymptotic behavior of the system for large t ≪ δ to derive g independent linear equations for the initial data Z0, ensuring unique reconstruction.
• Leverage the density of rational conserved quantities and the cardinality equivalence between the isolevel set of the pBBS and the tropical Jacobian to prove surjectivity of the map ισ.

Experimental results

Research questions

• RQ1 Is the generic isolevel set of the ultra-discrete periodic Toda lattice isomorphic to the tropical Jacobian of its tropical spectral curve?
• RQ2 Can the theta function solution previously proposed for the UD-pToda be shown to be complete via this isomorphism?
• RQ3 How can the initial value problem of the ultra-discrete periodic Toda lattice be systematically solved using tropical geometric structures?
• RQ4 What is the precise correspondence between the dynamics of the UD-pToda and the lattice points of the tropical Jacobian under generic conditions?

Key findings

• The generic isolevel set TC of the ultra-discrete periodic Toda lattice is isomorphic to the tropical Jacobian J(K) of the associated tropical spectral curve, under the generic condition C−1 > 2C0 and Ci + Ci+2 > 2Ci+1 for i = 0,…,g−2, and Cg−1 > 2Cg.
• The isomorphism is realized via a continuous, bijective map ισ from the tropical Jacobian to the isolevel set, proven by injectivity and surjectivity arguments using dense subsets and rational approximations.
• The cardinality of the isolevel set (TC)N for rational conserved quantities C is shown to be (g+1) times the number of lattice points in the tropical Jacobian, confirming the isomorphism at the level of integer points.
• The initial value problem is solved by reconstructing the initial data Z0 from the asymptotic behavior of the system at large times t ≪ δ, using g independent linear equations derived from the region Dm to which the point Znt belongs.
• The solution is expressed in terms of the quasi-periodic function Tnt, which satisfies a Hirota-type bilinear equation, and the phase space variables are recovered via the transformation (1.8).
• The proof relies on the equivalence between the isolevel set of the pBBS and the tropical Jacobian, extended to the UD-pToda via ultra-discretization and tropical geometry.