Numerical Inverse Scattering for the Toda Lattice

TL;DR

This paper introduces a numerical method to compute the inverse scattering transform for the Toda lattice by solving a Riemann-Hilbert problem efficiently, enabling analysis across all time regimes without time-stepping.

Contribution

It develops a novel numerical approach for the Toda lattice's inverse scattering transform using Riemann-Hilbert problem deformations, allowing efficient computation at any point in the domain.

Findings

Achieves $ ext{O}(1)$ complexity for evaluating solutions at arbitrary points.

Enables computation of long-time asymptotics where rigorous results are unavailable.

Provides a practical numerical framework for inverse scattering in integrable systems.

Abstract

We present a method to compute the inverse scattering transform (IST) for the famed Toda lattice by solving the associated Riemann--Hilbert (RH) problem numerically. Deformations for the RH problem are incorporated so that the IST can be evaluated in $\mathcal O(1)$ operations for arbitrary points in the $(n,t)$-domain, including short- and long-time regimes. No time-stepping is required to compute the solution because $(n,t)$ appear as parameters in the associated RH problem. The solution of the Toda lattice is computed in long-time asymptotic regions where the asymptotics are not known rigorously.