Computation of Large-Genus Solutions of the Korteweg-de Vries Equation

TL;DR

This paper presents a numerical method for computing large-genus solutions of the Korteweg-de Vries equation, enabling stable calculations of complex solutions for initial-value and dressing scenarios.

Contribution

It introduces a weighted Chebyshev basis approach and extends techniques for large-genus spectral data computation, improving stability and applicability.

Findings

Successfully computed large-genus solutions for various initial data

Demonstrated dispersive quantization for 'box' initial data

Produced new classes of potentials via large genus limits

Abstract

We consider the numerical computation of finite-genus solutions of the Korteweg-de Vries equation when the genus is large. Our method applies both to the initial-value problem when spectral data can be computed and to dressing scenarios when spectral data is specified arbitrarily. In order to compute large genus solutions, we employ a weighted Chebyshev basis to solve an associated singular integral equation. We also extend previous work to compute period matrices and the Abel map when the genus is large, maintaining numerical stability. We demonstrate our method on four different classes of solutions. Specifically, we demonstrate dispersive quantization for "box" initial data and demonstrate how a large genus limit can be taken to produce a new class of potentials.