Solutions to the SU(N) self-dual Yang-Mills equation

In this paper we aim to derive solutions for the SU(N) self-dual Yang-Mills (SDYM) equation with arbitrary 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(N) SDYM equation and also analyze the physical significance of some solutions, i.e. being Hermitian, positive-definite and of determinant being one.& COPY; 2023 Elsevier B.V. All rights reserved.

Automated summary

In this paper, solutions for the SU(N) self-dual Yang-Mills (SDYM) equation with arbitrary N are derived. 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 Cauchy matrix schemes for the Kadomtsev-Petviashvili hierarchy and the Ablowitz-Kaup-Newell-Segur hierarchy, respectively. The possible reductions that lead to the SU(N) SDYM equation are investigated in each Cauchy matrix scheme, and the physical significance of some solutions, such as being Hermitian, positive-definite, and having a determinant of one, is analyzed. ©2023 Elsevier B.V. All rights reserved.