Finite genus solutions to the lattice Schwarzian Korteweg-de Vries equation

TL;DR

This paper develops a new discrete Lax pair for the lattice Schwarzian KdV equation using Darboux transformations, enabling the calculation of finite genus solutions via Riemann surface methods.

Contribution

It introduces a novel Lax pair for lSKdV and applies the discrete Liouville-Arnold theorem to derive finite genus solutions.

Findings

New discrete Lax pair for lSKdV

Finite genus solutions computed via Riemann surfaces

Connection between integrable systems and algebraic geometry

Abstract

Based on integrable Hamiltonian systems related to the derivative Schwarzian Korteweg-de Vries (SKdV) equation, a novel discrete Lax pair for the lattice SKdV (lSKdV) equation is given by two copies of a Darboux transformation which can be used to derive an integrable symplectic correspondence. Resorting to the discrete version of Liouville-Arnold theorem, finite genus solutions to the lSKdV equation are calculated through Riemann surface method.