Numerical computation of the finite-genus solutions of the Korteweg-de Vries equation via Riemann-Hilbert problems

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Abstract

In this letter we describe how to compute the finite-genus solutions of the Korteweg-de Vries equation using a Riemann-Hilbert problem that is satisfied by the Baker-Akhiezer function corresponding to a Schrödinger operator with finite-gap spectrum. The recovery of the corresponding finite-genus solution is performed using the asymptotics of the Baker-Akhiezer function. This method has the benefit that the space and time dependence of the Baker-Akhiezer function appear in an explicit, linear and computable way. We make use of recent advances in the numerical solution of Riemann-Hilbert problems to produce an efficient and uniformly accurate numerical method for computing all finite-genus solutions of the KdV equation. © 2012 Elsevier Ltd. All rights reserved.

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Trogdon, T., & Deconinck, B. (2013). Numerical computation of the finite-genus solutions of the Korteweg-de Vries equation via Riemann-Hilbert problems. Applied Mathematics Letters, 26(1), 5–9. https://doi.org/10.1016/j.aml.2012.07.019

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