A purely dispersive solution of the Toda lattice computed at $t=10000$.

# Numerical Inverse Scattering Transforms

A modern approach is to consider solutions of $(1+1)$-dimensional nonlinear integrable models of wave propagation as nonlinear special functions that have their Riemann-Hilbert problem representations just as classical special functions (e.g. Airy, Bessel functions) have their integral representations. Riemann-Hilbert problems provide non-commutative analogues of integral representations.

## Goal

To compute physically interesting solutions of various integrable systems available using their Riemann-Hilbert problem representations and make these solutions computationally available to researchers in nonlinear waves community via a Julia package in an asymptotically robust way.

## Background

In 2012, Sheehan Olver developed a numerical framework to solve Riemann-Hilbert problems in context of computation of Painlevé-II transcendents ( RHPACKAGE). Following this development, Bernard Deconinck- Sheehan Olver- Thomas Trogdon introduced a numerical framework to compute solutions of nonlinear integrable $(1+1)$-dimensional dispersive partial differential equations, starting with the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations in 2014, and then focusing and defocusing nonlinear Schrödinger (NLS) equations. The numerical methodology developed is based on solving the Riemann-Hilbert problem that represents the solution of the PDE for a given $(x,t)$ through the associated inverse scattering transform. With the implementation of the Deift-Zhou method of nonlinear steepest descent, this framework can compute solutions of such dispersive PDEs in the entire $(x,t)$-plane with remarkable accuracy, in particular for arbitrarily large values of $t$, in an asymptotically robust way. All of these modules are publicly available under the Mathematica package ISTPACKAGE written and maintained by Thomas Trogdon.

I joined the workforce in 2015 to compute solutions of the initial value problem for the doubly-infinite Toda lattice on the entire $(n,t)$-domain, including long-time asymptotic regions that were not previously studied (see Numerical Inverse Scattering Transform for the Toda Lattice) and I am one of the contributors of the ISTPACKAGE since then.

As variables $(x,t)$ of the PDE (or $(n,t)$ of a P$\Delta$E) appear only as explicit parameters in the Riemann-Hilbert problem,

• no time-stepping or spatial discretization is necessary to compute the solution of the Cauchy problem under study,
• presence of high-frequency oscillations in the spatial variable is no longer a numerical challenge; in fact, such aspects become completely irrelevant,
• the numerical procedure is immediately parallelizable (a so-called “embarrassingly parallel” computation).

## Current work

Current work is on

• computing dispersive shock wave solutions (resulting from initial data with a step-like boundary values at infinity) of the Korteweg-de Vries (KdV) equation in arbitrarily large time-scales,
• investigating and computing presence of solitons trapped in a rarefaction fan solution of the KdV equation.

This project involves collaboration with Thomas Trogdon.

##### Deniz Bilman
###### Assistant Professor

Assistant Professor of Mathematics. Research interests include nonlinear waves, Riemann-Hilbert problems.