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 construct a matrix that can be reduced SDYM equation. It is shown these generated from two different Sylvester equations, which correspond Cauchy schemes (matrix) Kadomtsev-Petviashvili hierarchy and Ablowitz-Kaup-...

Based on the higher-order restricted flows, first type of integrable deformed fourth-order matrix NLS equations, that is, equations with self-consistent sources (FMNLSSCS), is derived. By virtue $${\bar{\partial }}$$ -dressing method, second called NLS–Maxwell–Bloch system (FMNLS-MB) presented. We prove equivalence FMNLSSCS and FMNLS-MB successfully. Furthermore, N-soliton solutions are explici...

This article makes use of simultaneous decomposition four quaternion matrixes to investigate some Sylvester-like matrix equation systems. We present useful necessary and sufficient conditions for the consistency system equations in terms equivalence form block matrixes. also derive general solution according partition coefficient As an application system, we practical a ϕ-Hermitian provide when...

This paper presents preconditioned Galerkin and minimal residual algorithms for the solution of Sylvester equations AX XB = C. Given two good preconditioner matricesM and N for matrices A and B, respectively, we solve the Sylvester equations MAXN MXBN =MCN. The algorithms use the Arnoldi process to generate orthonormal bases of certain Krylov subspaces and simultaneously reduce the order of Syl...

A spectral method for solving linear partial differential equations (PDEs) with variable coefficients and general boundary conditions defined on rectangular domains is described, based on separable representations of partial differential operators and the one-dimensional ultraspherical spectral method. If a partial differential operator is of splitting rank 2, such as the operator associated wi...