A direct method in noncommutative integrable systems

TL;DR

This paper develops a direct method for deriving noncommutative integrable equations using quasi-determinants, extending classical techniques to the noncommutative setting and producing explicit matrix soliton solutions.

Contribution

It introduces a constructive framework based on quasi-determinants for deriving NC integrable equations directly, including reductions and explicit solutions.

Findings

Derived NC KP and NC DJKM equations from quasi-determinant identities.

Obtained NC KdV and Boussinesq equations via flow constraints.

Produced explicit matrix-valued soliton solutions.

Abstract

We present a constructive framework for deriving noncommutative (NC) integrable equations directly from quasi-determinant solutions. Building upon the quasi-Wronskian structure, we extend the classical direct method to the NC setting, where standard determinant identities are replaced by algebraic relations intrinsic to quasi-determinants. By analyzing derivative identities satisfied by quasi-determinants, we recover the NC Kadomtsev-Petviashvili (ncKP) and Date-Jimbo-Kashiwara-Miwa (ncDJKM) equations by cancellations of nonlinear terms. Furthermore, by imposing flow constraints on the seed functions, we derive NC reductions such as the ncKdV and ncBoussinesq equations and obtain explicit matrix-valued soliton solutions. Our results highlight the quasi-determinant as a fundamental algebraic structure underpinning NC $\tau$-function structure.