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Creators/Authors contains: "Ballew, Cade"

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  1. ; ; (, Journal of Nonlinear Waves)
    We consider the problem of computing a class of soliton gas primitive potentials for the Korteweg-de Vries equation that arise from the accumulation of solitons on an infinite interval in the physical domain, extending to $$-\infty$$. This accumulation results in an associated Riemann-Hilbert Problem (RHP) on a number of disjoint intervals. In the case where the jump matrices have specific square-root behaviour, we describe an efficient and accurate numerical method to solve this RHP and extract the potential. The keys to the method are, first, the deformation of the RHP, making numerical use of the so-called $$g$$-function, and, second, the incorporation of endpoint singularities into the chosen basis to discretize and solve the associated singular integral equation. 
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  2. ; (, Studies in Applied Mathematics)
    Abstract We develop a numerical method for computing with orthogonal polynomials that are orthogonal on multiple, disjoint intervals for which analytical formulae are currently unknown. Our approach exploits the Fokas–Its–Kitaev Riemann–Hilbert representation of the orthogonal polynomials to produce an method to compute the firstNrecurrence coefficients. The method can also be used for pointwise evaluation of the polynomials and their Cauchy transforms throughout the complex plane. The method encodes the singularity behavior of weight functions using weighted Cauchy integrals of Chebyshev polynomials. This greatly improves the efficiency of the method, outperforming other available techniques. We demonstrate the fast convergence of our method and present applications to integrable systems and approximation theory. 
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