Solutions to the SU([formula omitted]) 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. • Solutions of the SU(N) SDYM equation with arbitrary N are constructed. • The SU(N) SDYM is formulated by the Cauchy matrix scheme of the KP hierarchy. • The SU(2N) SDYM is formulated by the Cauchy matrix scheme of the AKNS hierarchy. • Some solutions of the SU(2N) SDYM obtained in the two schemes are analyzed. [ABSTRACT FROM AUTHOR]