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1. 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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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. Generalized matrix exponential solutions to the AKNS hierarchy

   Zhang, Jian-bing; Gu, Canyuan; Ma, Wen-Xiu (February 2018, Advances in mathematical physics)

   Generalized matrix exponential solutions to the AKNS equation are obtained by the inverse scattering transformation (IST). The resulting solutions involve six matrices, which satisfy the coupled Sylvester equations. Several kinds of explicit solutions including soliton, complexiton, and Matveev solutions are deduced from the generalized matrix exponential solutions by choosing different kinds of the six involved matrices. Generalized matrix exponential solutions to a general integrable equation of the AKNS hierarchy are also derived. It is shown that the general equation and its matrix exponential solutions share the same linear structure. 

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4. A Riemann–Hilbert Problem Approach to Infinite Gap Hill's Operators and the Korteweg–de Vries Equation

   https://doi.org/10.1093/imrn/rnz156  

   T-R McLaughlin, Kenneth; Nabelek, Patrik V. (November 2019, International Mathematics Research Notices)

   Abstract We formulate the inverse spectral theory of infinite gap Hill’s operators with bounded periodic potentials as a Riemann–Hilbert problem on a typically infinite collection of spectral bands and gaps. We establish a uniqueness theorem for this Riemann–Hilbert problem, which provides a new route to establishing unique determination of periodic potentials from spectral data. As the potentials evolve according to the Korteweg–de Vries Equation (KdV) equation, we use integrability to derive an associated Riemann–Hilbert problem with explicit time dependence. Basic principles from the theory of Riemann–Hilbert problems yield a new characterization of spectra for periodic potentials in terms of the existence of a solution to a scalar Riemann–Hilbert problem, and we derive a similar condition on the spectrum for the temporal periodicity for an evolution under the KdV equation. 

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5. Scattering and inverse scattering for the AKNS system: A rational function approach

   https://doi.org/10.1111/sapm.12434  

   Trogdon, Thomas (August 2021, Studies in Applied Mathematics)

   Abstract We consider the use of rational basis functions to compute the scattering and inverse scattering transforms associated with the AKNS (Ablowitz–Kaup–Newell–Segur) system. The proposed numerical forward scattering transform computes the solution of the AKNS system that is valid on the entire real axis and thereby computes a reflection coefficient at a point by solving a single linear system. The proposed numerical inverse scattering transform makes use of a novel improvement in the rational function approach to the oscillatory Cauchy operator, enabling the efficient solution of certain Riemann–Hilbert problems without contour deformations. The latter development enables access to high‐precision computations and this is demonstrated on the inverse scattering transform for the one‐dimensional Schrödinger operator with a potential. 

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