The Cauchy matrix approach is developed to solve the boldSUfalse(2false)$\mathbf {SU}(2)$ self‐dual Yang–Mills (SDYM) equation. Starting from a Sylvester matrix equation coupled with certain dispersion relations for an infinite number of coordinates, we derive some new relations that give rise to the SDYM equation under Yang's formulation. By imposing further constraints on complex independent variables, a broad class of explicit solutions of the equation under Yang's formulation are obtained.

Развит прямой метод - метод матриц Коши - для построения матричных решений некоммутативных солитонных уравнений. Метод основан на уравнении Сильвестра, при этом решения представлены без использования квазидетерминантов. В качестве примера, демонстрирующего метод, используется матричное уравнение Кадомцева-Петвиашвили с самосогласованными источниками. С помощью редукции получены явные решения матричной модели Мельникова для длинно-коротковолнового взаимодействия.

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.

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We show that when KP (Kadomtsev–Petviashvili) τ functions allow special symmetries, the discrete BKP equation can be expressed as a linear combination of the discrete AKP equation and its reflected symmetric forms. Thus the discrete AKP and BKP equations can share the same τ functions with these symmetries. Such a connection is extended to 4 dimensional (i.e. higher order) discrete AKP and BKP equations in the corresponding discrete hierarchies. Various explicit forms of such τ functions, including Hirota’s form, Gramian, Casoratian and polynomial, are given. Symmetric τ functions of Cauchy matrix form that are composed of Weierstrass σ functions are investigated. As a result we obtain a discrete BKP equation with elliptic coefficients.