Nonlinear Spectral Singularities for Localized Nonlinearities

TL;DR

This paper introduces nonlinear spectral singularities in Schrödinger operators with confined nonlinearities, revealing amplitude-dependent resonance effects that produce amplified waves with specific profiles.

Contribution

It extends the concept of spectral singularities to nonlinear operators, showing they remain symmetry-preserving but become amplitude-dependent, with implications for nonlinear wave phenomena.

Findings

Spectral singularities are preserved under nonlinearities.

Nonlinear spectral singularities depend on wave amplitude.

Resonance effects lead to amplified waves with specific profiles.

Abstract

We introduce a notion of spectral singularity that applies for a general class of nonlinear Schreodinger operators involving a confined nonlinearity. The presence of the nonlinearity does not break the parity-reflection symmetry of spectral singularities but makes them amplitude-dependent. Nonlinear spectral singularities are, therefore, associated with a resonance effect that produces amplified waves with a specific amplitude-wavelength profile. We explore the consequences of this phenomenon for a complex delta-function potential that is subject to a general confined nonlinearity.