Solution of the genaralized periodic discrete Toda equation

TL;DR

This paper explores the generalized periodic discrete Toda equation, linking it to algebraic geometry, and provides explicit theta function solutions by linearizing its time evolution on an algebraic variety.

Contribution

It introduces a novel algebraic geometric approach to solving the generalized periodic discrete Toda equation with explicit theta function solutions.

Findings

Time evolution is linearized on an algebraic variety.

Explicit theta function solutions are constructed.

The approach extends understanding of integrable systems.

Abstract

A box-ball system with more than one kind of balls is obtained by the generalized periodic discrete Toda equation (pd Toda eq.). We study the pd Toda equation in view of algebraic geometry. The time evolution of pd Toda eq. is linearized on an algebraic variety, and theta function solutions are obtained.