An Exactly Solvable Model for Nonlinear Resonant Scattering

TL;DR

This paper presents an exactly solvable nonlinear model for resonant scattering phenomena, revealing power laws and stability properties of harmonic solutions near resonance, with implications for understanding Fano resonances.

Contribution

It introduces a simple, exactly solvable model capturing nonlinear resonant scattering effects and rigorously analyzes the stability and asymptotic behavior of harmonic solutions.

Findings

Power law <mu> ~ C<gamma>^4 for small coupling and nonlinearity

Widening of frequency intervals of harmonic solutions as parameters vary

Stable and unstable regimes identified for different harmonic responses

Abstract

This work analyzes the effects of cubic nonlinearities on certain resonant scattering anomalies associated with the dissolution of an embedded eigenvalue of a linear scattering system. These sharp peak-dip anomalies in the frequency domain are often called Fano resonances. We study a simple model that incorporates the essential features of this kind of resonance. It features a linear scatterer attached to a transmission line with a point-mass defect and coupled to a nonlinear oscillator. We prove two power laws in the small coupling <gamma> \to 0 and small nonlinearity <mu> \to 0 regime. The asymptotic relation <mu> ~ C<gamma>^4 characterizes the emergence of a small frequency interval of triple harmonic solutions near the resonant frequency of the oscillator. As the nonlinearity grows or the coupling diminishes, this interval widens and, at the relation <mu> ~ C<gamma>^2, merges with another evolving frequency interval of triple harmonic solutions that extends to infinity. Our model allows rigorous computation of stability in the small <mu> and <gamma> limit. In the regime of triple harmonic solutions, those with largest and smallest response of the oscillator are linearly stable and the solution with intermediate response is unstable.