Perfectly absorbed and emitted currents by complex potentials in nonlinear media

TL;DR

This paper explores nonlinear spectral singularities in complex potentials within nonlinear media, extending the class of solutions to include inhomogeneous amplitude profiles and analyzing their bifurcation from linear limits.

Contribution

It introduces a broader class of nonlinear spectral singularities with inhomogeneous amplitudes and examines how these solutions relate to linear spectral singularities through potential deformation.

Findings

Nonlinear currents can be bifurcated from linear spectral singularities.

Some nonlinear solutions cannot be reduced to the linear limit.

Deformation of the potential is necessary for continuation from the linear case.

Abstract

Recently it was demonstrated that the concept of a spectral singularity (SS) can be generalized to waves propagating in nonlinear media, like matter waves or electromagnetic waves in Kerr media. The corresponding solutions represent nonlinear currents sustained by a localized linear complex potential in a nonlinear Schr\"odinger equation. A key feature allowing the nonlinear generalization of a SS is a possibility to reduce a nonlinear current to the linear limit, where a SS has the unambiguous definition. In the meantime, known examples of nonlinear modes bifurcating from linear spectral singularities are few and belong to the specific class of constant-amplitude waves. Here we propose to extend the class of nonlinear SSs by incorporating solutions whose amplitudes are inhomogeneous. We show that the continuation from the linear limit requires a deformation of the complex potential, and this deformation is not unique. Examples include the deformation preserving the gain-and-loss distribution and the deformation preserving geometry of the potential. For the case example of a rectangular potential, we demonstrate that the nonlinear currents can be divided into two types: solutions of the first type bifurcate from the linear spectral singularities, and solutions of the second type cannot be reduced to the linear limit.