Near integrability of kink lattice with higher order interactions

TL;DR

This paper investigates the near integrability of a kink lattice with higher order interactions in 1+1 dimensional field theory, revealing its relation to Toda lattices and analyzing its algebraic structure and Hamiltonians.

Contribution

It demonstrates that the kink lattice with higher order corrections is a near integrable system related to the Toda lattice, providing explicit coefficients and algebraic relations.

Findings

Kink lattice reduces to Toda lattice at lowest order

Higher order corrections lead to a near integrable system

Non-integrability is shown through algebraic relations of Flaschka's variables

Abstract

In the paper, we make use of Manton's analytical method to investigate the force between kink and the anti-kink with large distance in $1+1$ dimensional field theory. The related potential has infinite order corrections of exponential pattern, and coefficients for each order are determined. These coefficients can also be obtained by solving the equation of the fluctuation around the vacuum. At the lowest order, the kink lattice represents the Toda lattice. With higher order correction terms, the kink lattice can represent one kind of the generic Toda lattice. With only two sites, the kink lattice is classically integrable. If the number of sites of the lattice is larger than two, the kink lattice is not integrable but a near integrable system. We take use of the Flaschka's variables to study the Lax pair of the kink lattice. These Flaschka's variables have interesting algebraic relations and the non-integrability can be manifested. We also discussed the higher Hamiltonians for the deformed open Toda lattice, which has a similar result as the ordinary deformed Toda.