On nonlinear scattering for quantum walks

TL;DR

This paper investigates the long-term behavior of nonlinear quantum walks with self-dependent coins, establishing scattering results, deriving inverse scattering formulas, and numerically observing soliton-like solutions, thus extending analytical techniques from nonlinear Schrödinger equations to quantum walks.

Contribution

It introduces the first application of space-time estimates to quantum walks with nonlinearities, demonstrating scattering and soliton-like phenomena in this context.

Findings

Proves scattering for nonlinear quantum walks.

Derives inverse scattering formulas in the weak nonlinear regime.

Numerical simulations show soliton-like solutions.

Abstract

We study large time behavior of quantum walks (QW) with self-dependent coin. In particular, we show scattering and derive the reproducing formula for inverse scattering in the weak nonlinear regime. The proof is based on space-time estimate of (linear) QW such as Strichartz estimate. Such argument is standard in the study of nonlinear Schr\"odinger equations but it seems to be the first time to be applied to QW. We also numerically study the dynamics of QW and observe soliton like solutions.