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arXiv · 2608.11892

Fast eight-order Pade schemes based on Chebyshev polynomials for direct Zakharov-Shabat problem

Sergey Medvedev·Igor Chekhovskoy·Irina Vaseva·Mikhail Fedoruk
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Abstract

In this work, we construct fast eighth-order Pade schemes for the direct Zakharov-Shabat scattering problem. The schemes are based on an eighth-order exponential integrator obtained from the Magnus expansion. A direct extension of the conventional fast Pade representation to the eighth-order case leads to insufficiently accurate fast variants in the considered tests. To overcome this difficulty, we reformulate the spectral dependence on a finite real interval using the Joukowski mapping and represent the local numerators and denominators in the Chebyshev polynomial basis. This makes it possible to construct the global transition matrix by a fast product tree while retaining a compact polynomial representation. Numerical experiments for chirped hyperbolic secant potentials with both signs of dispersion show that the proposed Chebyshev-based fast schemes substantially improve the accuracy of the corresponding direct fast variants and can be used for efficient computation of the continuous nonlinear spectrum.

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Sergey Medvedev, Igor Chekhovskoy, Irina Vaseva, Mikhail Fedoruk. 2026-08-12. Fast eight-order Pade schemes based on Chebyshev polynomials for direct Zakharov-Shabat problem. https://arxiv.org/abs/2608.11892

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