On scattering for NLS: rigidity properties and numerical simulations via the lens transform

TL;DR

This paper introduces a novel numerical method using the lens transform to efficiently compute the scattering operator for the nonlinear Schrödinger equation, supported by theoretical identities and numerical experiments.

Contribution

It presents the first application of the lens transform in numerical simulations of scattering, providing new identities and exploring regimes beyond current analytical results.

Findings

Validated the numerical method against known analytical properties.

Explored long-range scattering regimes and formulated new conjectures.

Demonstrated the method's efficiency and reliability in various regimes.

Abstract

We analyse the scattering operator associated with the defocusing nonlinear Schr{\"o}dinger equation which captures the evolution of solutions over an infinite time-interval under the nonlinear flow of this equation. The asymptotic nature of the scattering operator (involving unbounded time) makes its computation particularly challenging. We overcome this by exploiting the space-time compactification provided by the lens transform, marking the first use of this technique in numerical simulations. This results in a highly efficient and reliable methodology for computing the scattering operator in various regimes. In developing this approach we introduce and prove several new identities and theoretical properties of the scattering operator. We support our construction with several numerical experiments which we show to agree with known analytical properties of the scattering operator, and also address the case of long-range scattering for the one-dimensional cubic Schr{\"o}dinger equation. Our simulations permit us to further explore regimes beyond current analytical understanding, and lead us to formulate new conjectures concerning fixed and rotating points of the operator, as well as its existence in the long-range setting for both defocusing and focusing cases.