Conserved quantities and generalized solutions of the ultradiscrete KdV equation

TL;DR

This paper develops generalized solutions, including negative solitons, for the ultradiscrete KdV equation using ultradiscretization and gauge transformations, and explores their conserved quantities.

Contribution

It introduces a method to construct generalized solutions for the ultradiscrete KdV equation, expanding the understanding of its solution space and conserved quantities.

Findings

Constructed negative soliton solutions.

Established conserved quantities similar to the box-ball system.

Linked ultradiscrete KdV solutions to discrete KdV via ultradiscretization.

Abstract

We construct generalized solutions to the ultradiscrete KdV equation, including the so-called negative solition solutions. The method is based on the ultradiscretization of soliton solutions to the discrete KdV equation with gauge transformation. The conserved quantities of the ultradiscrete KdV equation are shown to be constructed in a similar way to those for the box-ball system.