Math @ Duke

Publications [#244166] of Stephanos Venakides
Papers Published
 Shipman, SP; Venakides, S, An exactly solvable model for nonlinear resonant scattering,
Nonlinearity, vol. 25 no. 9
(September, 2012),
pp. 24732501, IOP Publishing, ISSN 09517715 (doi:10.1088/09517715/25/9/2473.) [doi]
(last updated on 2019/04/20)
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 peakdip 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 pointmass 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→Cγ4characterizes 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. © 2012 IOP Publishing Ltd & London Mathematical Society.


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