Persistence of approximate integrability near the massive Thirring limit

Determine how close the parameter p in the scalar-scalar plus (1/p) vector-vector nonlinear Dirac model can be to zero while retaining observable effects of integrability, specifically by characterizing how well the formerly conserved quantities of the integrable massive Thirring model remain constant for nonzero p.

Background

For κ=1, taking p→0 in the scalar-scalar plus (1/p) vector-vector model yields the integrable massive Thirring model. The paper asks whether the integrable structure has measurable or mathematically characterizable remnants when p is small but nonzero, so that the model is a perturbation of the massive Thirring model rather than exactly integrable.

The unresolved issue concerns the behavior of quantities that are exactly conserved in the integrable limit: whether, and in what quantitative sense, their constancy persists under a small nonintegrable vector-vector perturbation. The authors relate this question to analogous work on discrete nonintegrable systems near the integrable Ablowitz–Ladik model.

References

For \kappa = 1 the VV model, i.e. the massive Thirring model (MTM), is known to be integrable. The obvious question is how close p should be equal to zero to see the effect of integrability (note that for \kappa = 1, and p\rightarrow 0 we obtain MTM). More precisely, how well is the constancy of the formerly conserved quantities is preserved in case p is very small but nonzero? Recently, such a question was posed and discussed in some detail in the context of a discrete non-integrable model vis a vis the integrable Ablowitz-Ladik model.

— Two-Parameter Family of Nonlinear Dirac Equations With Scalar-Scalar plus Vector-Vector Interactions  (2609.18170 - Khare et al., 16 Sep 2026) in Section V, Conclusions and Open Problems, item 5