Nonlinear Scattering and Its Transfer Matrix Formulation in One Dimension

TL;DR

This paper develops a systematic transfer matrix approach for nonlinear scattering in one dimension, enabling analysis of spectral singularities, invisibility, and nonreciprocal transmission in nonlinear systems.

Contribution

It introduces a nonlinear transfer matrix with a composition property and applies it to analyze nonlinear scattering phenomena and potentials.

Findings

Established a nonlinear transfer matrix with composition property

Characterized spectral singularities and invisibility in nonlinear systems

Applied framework to nonlinear delta-function potentials

Abstract

We present a systematic formulation of scattering theory for nonlinear interactions in one dimension and develop a nonlinear generalization of the transfer matrix that has a composition property similar to its linear analog's. We offer alternative characterizations of spectral singularities, unidirectional reflectionlessness and invisibility, and nonreciprocal transmission for nonlinear scattering systems, and examine the application of our general results in addressing the scattering problem for nonlinear single- and double-$\delta$-function potentials.