Emergence of Reflectionless Scattering from Linearizations of Integrable PDEs around Solitons

TL;DR

This paper demonstrates that linearizations of certain integrable PDEs around solitons exhibit reflectionless scattering at all energies, highlighting a connection between integrability and transparency properties.

Contribution

It establishes a link between integrability of PDEs and reflectionless scattering in their linearized forms, supported by multiple examples including both S- and C-integrable systems.

Findings

Linearized integrable PDEs show reflectionless scattering at all energies.

The transparency property extends from solitons to their linearizations in integrable systems.

Non-integrable systems do not exhibit reflectionless scattering in their linearizations.

Abstract

We present four examples of integrable partial differential equations (PDEs) of mathematical physics that---when linearized around a stationary soliton---exhibit scattering without reflection at {\it all} energies. Starting from the most well-known and the most empirically relevant phenomenon of the transparency of one-dimensional bright bosonic solitons to Bogoliubov excitations, we proceed to the sine-Gordon, Korteweg-de Vries, and Liouville's equation whose stationary solitons also support our assertion. The proposed connection between integrability and reflectionless scattering seems to span at least two distinct paradigms of integrability: S-integrability in the first three cases, and C-integrability in the last one. We argue that the transparency of linearized integrable PDEs is necessary to ensure that they can support the transparency of stationary solitons in the original integrable PDEs. As contrasting cases, the analysis is further extended to cover two non-integrable systems: a sawtooth-Gordon and a $\phi^4$ model.