The generalized periodic ultradiscrete KdV equation and its background solutions

TL;DR

This paper studies the ultradiscrete KdV equation with periodic boundary conditions, providing criteria for initial states, conserved quantities, and constructing background solutions using Jacobi theta functions.

Contribution

It introduces a criterion for periodic boundary conditions with arbitrary integer parameters and constructs background solutions via Jacobi theta functions.

Findings

Established a criterion for periodic boundary conditions with arbitrary integers.

Constructed conserved quantities for the periodic ultradiscrete KdV system.

Developed background solutions using Jacobi theta functions.

Abstract

We investigate the ultradiscrete KdV equation with periodic boundary conditions where the two parameters (capacity of the boxes and that of the carrier) are arbitrary integers. We give a criterion to allow a periodic boundary condition when initial states take arbitrary integer values. Conserved quantities are constructed for the periodic systems. Construction of background solutions of the periodic ultradiscrete KdV equation from the Jacobi theta function is also presented.