Modified open Toda chain and quasi-integrability

TL;DR

This work investigates a quasi-integrable deformation of the three-particle open Toda chain by adding a translation-invariant three-body term. It shows that while the Toda hierarchy is not fully preserved, the energy $H$ and momentum $P$ remain exactly conserved, and a higher-order invariant $I_3$ becomes asymptotically conserved in a symmetric sector thanks to a discrete time-reversal/parity symmetry, with anomaly ${\cal A}_3(t)$ governing the deficit. The anomaly is derived exactly and analyzed both analytically and numerically, including a perturbative expansion in the deformation parameter $\epsilon$, which reveals the leading nonzero contribution at $O(\epsilon^3)$ and its vanishing upon symmetric initial data. The results provide a minimal mechanical model of quasi-integrability, illustrating how non-integrable deformations can preserve key structural features and offering a platform to explore the onset of quasi-integrability in few-body systems and potential generalizations to larger $N$.

Abstract

We present a study of a quasi-integrable deformation of the three-particle open Toda chain, constructed by introducing a translation-invariant three-body interaction terms. Although this modification explicitly breaks the exact integrability of the standard Toda model, it retains fundamental structural properties, including energy and momentum conservation. Furthermore, we show that under a specific time-reflection and discrete symmetry among the chain coordinates, the system admits a quasi-conserved higher-order integral. Through analytic and numerical analysis of the deformed dynamics, we demonstrate the emergence and long-time persistence of quasi-conserved quantities, thereby establishing a controlled realization of quasi-integrability in a minimal nonlinear chain. Given the central role of integrable systems in elucidating the dynamics of classical and quantum models, this framework provides a concrete setting to investigate the mechanisms underlying the gradual breakdown of integrability and the onset of quasi-integrability in few-body systems.

Figures (4)

• Figure 1: (color online) A solution of the $N=3$ Toda chain coordinates $q_1(t)$ (red), $q_2(t)$ (blue) and $q_3(t)$ (green). Notice that the symmetry (\ref{['sym1']}) is realized.
• Figure 2: (color online) The $N=3$ modified Toda chain coordinates $q_1(t)$ (red), $q_2(t)$ (blue) and $q_3(t)$ (green) up to the first order contribution for $\epsilon=0.5$. Notice that the symmetry (\ref{['sym1']}) is realized.
• Figure 3: (color online) Plots of energy $E(t)$ and the third charge $I_3$ for $\epsilon =0.12$ evaluated for the coordinates $q_i(t) = q_i^0+ \epsilon q_i^1,\, i=1,2,3$. Notice the asymptotically constant $I_3(t=\pm t_0) = 0.5$ behavior for large $|t|= t_0 > 2.5$.
• Figure 4: (color online) The numerical solution for $N=3$ modified Toda chain coordinates $q_1(t)$ (red), $q_2(t)$ (blue) and $q_3(t)$ (green) for $\epsilon=0.75$. Notice that the symmetry (\ref{['sym1']}) of the coordinates is realized. The dark line shows the anomaly ${\cal A}_3(t)$ with the odd symmetry (\ref{['odd1']}).