Lax pair and Darboux transformation of noncommutative U(N) principal chiral model

TL;DR

This paper develops a noncommutative extension of the Lax formalism, Bäcklund transformations, and Darboux transformations for the U(N) principal chiral model, enabling analysis of integrability in noncommutative geometry.

Contribution

It introduces a noncommutative Lax formalism, Bäcklund transformations, and Darboux transformations for the U(N) principal chiral model, expanding integrability tools to noncommutative settings.

Findings

Derived a noncommutative Lax pair for the model

Constructed a noncommutative Bäcklund transformation

Established a noncommutative Darboux transformation

Abstract

We present a noncommutative generalization of Lax formalism of U(N) principal chiral model in terms of a one-parameter family of flat connections. The Lax formalism is further used to derive a set of parametric noncommutative B\"{a}cklund transformation and an infinite set of conserved quantities. From the Lax pair, we derive a noncommutative version of the Darboux transformation of the model.