Numerical Inverse Scattering for the Toda Lattice

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.

💡 Research Summary

This paper presents a complete numerical framework for computing the inverse scattering transform (IST) of the Toda lattice, a prototypical infinite‑dimensional integrable system. The authors start by recalling the Lax pair formulation (L, P) of the Toda lattice, which guarantees integrability and leads to a scattering theory for the associated Jacobi operator L. Assuming exponentially decaying initial data, the spectrum of L consists of an absolutely continuous part on