On the singularity structure of a discrete modified-Korteweg-deVries equation

TL;DR

This paper investigates the singularity structures of a modified lattice KdV equation, revealing three families of singularities and their interactions, which are similar to those in the classical lattice KdV equation, and explores their implications for integrability.

Contribution

It characterizes the singularity families of the modified lattice KdV equation and introduces a symbolic framework to describe their interactions, extending the concept of taishi singularities beyond the lattice KdV.

Findings

Identifies three families of singularities in the modified lattice KdV equation.

Shows that taishi-type singularities are present and interact richly with other singularities.

Provides a symbolic representation of singularity dynamics related to the ultradiscrete mKdV system.

Abstract

We study the singularities of a modified lattice Korteweg-deVries (KdV) equation and show that it admits three families of singularities, with analogous properties to those found in the lattice KdV equation. The first family consists of localised singularities which can occupy an arbitrarily large domain but which are, nevertheless, always confined. The second family consists of one or more lines extending all the way from the south-west to the north-east on the plane, involving a single finite value that depends on the parameter that appears in the equation. We argue that the infinite extent of this singularity is not incompatible with the confinement property or with the integrability of the equation. The third family consists of horizontal strips in which the product of values on vertically adjacent lattice sites is equal to 1. In the case of the lattice KdV equation this type of singularity was dubbed `taishi'. The taishi for the modified lattice KdV equation can interact with singularities of the other two families, giving rise to very rich {and quite intricate} singularity structures. Nonetheless, these interactions can be described in a compact way through the formulation of a symbolic representation of the dynamics that is similar to, but in a sense, simpler than that for the KdV case. We give an interpretation of this symbolic representation in terms of a Box&Ball system related to the ultradiscrete mKdV equation. These results show that taishi-type singularities are not limited to the lattice KdV equation, but might very well be a general feature of integrable lattice equations with deep connections to other facets of their integrability.