Dispersionless limit of the noncommutative potential KP hierarchy and solutions of the pseudodual chiral model in 2+1 dimensions

TL;DR

This paper develops a dispersionless limit for the noncommutative potential KP hierarchy, leading to a new method for constructing exact solutions of the pseudodual chiral model in 2+1 dimensions, including lump configurations.

Contribution

It introduces a novel dispersionless limit for the noncommutative potential KP hierarchy and a solution-generating method for the pseudodual chiral model.

Findings

Constructed classes of exact solutions for the su(m) pseudodual chiral model.

Extended solution generation techniques from the matrix KP hierarchy to the dispersionless case.

Demonstrated lump configurations in the solutions of the 2+1 dimensional model.

Abstract

The usual dispersionless limit of the KP hierarchy does not work in the case where the dependent variable has values in a noncommutative (e.g. matrix) algebra. Passing over to the potential KP hierarchy, there is a corresponding scaling limit in the noncommutative case, which turns out to be the hierarchy of a `pseudodual chiral model' in 2+1 dimensions (`pseudodual' to a hierarchy extending Ward's (modified) integrable chiral model). Applying the scaling procedure to a method generating exact solutions of a matrix (potential) KP hierarchy from solutions of a matrix linear heat hierarchy, leads to a corresponding method that generates exact solutions of the matrix dispersionless potential KP hierarchy, i.e. the pseudodual chiral model hierarchy. We use this result to construct classes of exact solutions of the su(m) pseudodual chiral model in 2+1 dimensions, including various multiple lump configurations.