An exactly solvable model for nonlinear resonant scattering

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 (gamma -> 0) and small nonlinearity (mu -> 0) regime. The asymptotic relation mu similar to 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 similar to 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. 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.