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An Exactly Solvable Model for Nonlinear Resonant Scattering
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 \to 0 and small nonlinearity \to 0 regime. The asymptotic
relation ~ C^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 ~ C^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 and
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
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