An exactly solvable model for nonlinear resonant scattering

Abstract

This work analyses 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 (γ → 0) and small nonlinearity (μ → 0) regime. The asymptotic relation μ ∼ Cγ4 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 μ ∼ Cγ2, merges with another evolving frequency interval of triple harmonic solutions that extends to infinity. Our model allows rigorous computation of stability in the small μ and γ limit. The regime of triple harmonic solutions exhibits bistability—those solutions with largest and smallest response of the oscillator are linearly stable and the solution with intermediate response is unstable.