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Mathematical Physics, math-ph,High Energy Physics - Theory, hep-th,Mathematics, Mathematical Physics, math.MP,Nonlinear Sciences, Exactly Solvable and Integrable Systems, nlin.SI
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.
