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1. Riemann-Hilbert problems and N-soliton solutions for a coupled mKdV system

   Ma, Wen-Xiu (October 2018, Journal of geometry and physics)

   A 3×3 matrix spectral problem is introduced and its associated AKNS integrable hierarchy with four components is generated. From this spectral problem, a kind of Riemann–Hilbert problems is formulated for a system of coupled mKdV equations in the resulting AKNS integrable hierarchy. N-soliton solutions to the coupled mKdV system are presented through a specific Riemann–Hilbert problem with an identity jump matrix. 

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2. Darboux transformations of integrable couplings and applications

   Ma, Wen-Xiu; Zhang, Yu-Juan (November 2017, Reviews in mathematical physics)

   A formulation of Darboux transformations is proposed for integrable couplings, based on non-semisimple matrix Lie algebras. Applications to a kind of integrable couplings of the AKNS equations are made, along with an explicit formula for the associated Bäcklund transformation. Exact one-soliton-like solutions are computed for the integrable couplings of the second- and third-order AKNS equations, and a type of reduction is created to generate integrable couplings and their one-soliton-like solutions for the NLS and MKdV equations. 

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3. Riemann–Hilbert approach for an initial-boundary value problem of the two-component modified Korteweg-de Vries equation on the half-line

   https://doi.org/https://doi.org/10.1016/j.amc.2018.03.049  

   Hu, Bei-Bei; Xia, Tie-Cheng; Ma, Wen-Xiu (September 2018, Applied mathematics and computation)

   In this work, we investigate the two-component modified Korteweg-de Vries (mKdV) equation, which is a complete integrable system, and accepts a generalization of 4 × 4 matrix Ablowitz–Kaup–Newell-Segur (AKNS)-type Lax pair. By using of the unified transform approach, the initial-boundary value (IBV) problem of the two-component mKdV equation associated with a 4 × 4 matrix Lax pair on the half-line will be analyzed. Supposing that the solution {u1(x, t), u2(x, t)} of the two-component mKdV equation exists, we will show that it can be expressed in terms of the unique solution of a 4 × 4 matrix Riemann–Hilbert problem formulated in the complex λ-plane. Moreover, we will prove that some spectral functions s(λ) and S(λ) are not independent of each other but meet the global relationship. 

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4. Some matrix properties preserved by generalized matrix functions

   https://doi.org/10.1515/spma-2019-0003  

   Benzi, Michele; Huang, Ru (January 2019, Special Matrices)

   null (Ed.)

   Abstract Generalized matrix functions were first introduced in [J. B. Hawkins and A. Ben-Israel, Linear and Multilinear Algebra, 1(2), 1973, pp. 163-171]. Recently, it has been recognized that these matrix functions arise in a number of applications, and various numerical methods have been proposed for their computation. The exploitation of structural properties, when present, can lead to more efficient and accurate algorithms. The main goal of this paper is to identify structural properties of matrices which are preserved by generalized matrix functions. In cases where a given property is not preserved in general, we provide conditions on the underlying scalar function under which the property of interest will be preserved by the corresponding generalized matrix function. 

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5. Enhanced alternating energy minimization methods for stochastic galerkin matrix equations

   https://doi.org/10.1007/s10543-021-00903-x  

   Lee, Kookjin; Elman, Howard C.; Powell, Catherine E.; Lee, Dongeun (January 2022, BIT numerical mathematics)

   In uncertainty quantification, it is commonly required to solve a forward model consisting of a partial differential equation (PDE) with a spatially varying uncertain coefficient that is represented as an affine function of a set of random variables, or parameters. Discretizing such models using stochastic Galerkin finite element methods (SGFEMs) leads to very high-dimensional discrete problems that can be cast as linear multi-term matrix equations (LMTMEs). We develop efficient computational methods for approximating solutions of such matrix equations in low rank. To do this, we follow an alternating energy minimization (AEM) framework, wherein the solution is represented as a product of two matrices, and approximations to each component are sought by solving certain minimization problems repeatedly. Inspired by proper generalized decomposition methods, the iterative solution algorithms we present are based on a rank-adaptive variant of AEM methods that successively computes a rank-one solution component at each step. We introduce and evaluate new enhancement procedures to improve the accuracy of the approximations these algorithms deliver. The efficiency and accuracy of the enhanced AEM methods is demonstrated through numerical experiments with LMTMEs associated with SGFEM discretizations of parameterized linear elliptic PDEs. 

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