Solutions to the SU($\mathcal{N}$) self-dual Yang-Mills equation

TL;DR

This paper derives solutions for the SU(N) self-dual Yang-Mills equation using noncommutative relations and matrix equations linked to integrable hierarchies, exploring reductions and physical properties of solutions.

Contribution

It introduces a novel approach connecting noncommutative relations with matrix equations from integrable hierarchies to solve the SU(N) SDYM equation.

Findings

Solutions derived from Sylvester equations linked to integrable hierarchies.

Reductions leading to specific SU(N) SDYM solutions analyzed.

Physical properties such as Hermiticity and positive-definiteness discussed.

Abstract

In this paper we aim to derive solutions for the SU($\mathcal{N}$) self-dual Yang-Mills (SDYM) equation with arbitrary $\mathcal{N}$. A set of noncommutative relations are introduced to construct a matrix equation that can be reduced to the SDYM equation. It is shown that these relations can be generated from two different Sylvester equations, which correspond to the two Cauchy matrix schemes for the (matrix) Kadomtsev-Petviashvili hierarchy and the (matrix) Ablowitz-Kaup-Newell-Segur hierarchy, respectively. In each Cauchy matrix scheme we investigate the possible reductions that can lead to the SU$(\mathcal{N})$ SDYM equation and also analyze the physical significance of some solutions, i.e. being Hermitian, positive-definite and of determinant being one.