Modulational instability of a wave scattered by a nonlinear center

We consider scattering of a quantum particle by a potential which includes a \ensuremath{\delta} function whose amplitude is nonlinear in the wave function. Solution of the scattering problem in this model is nonunique in a certain interval of amplitudes of the incident wave. We demonstrate that the nonlinearity gives rise to an oscillatory instability of the scattering solutions, which is a localized version of the well-known modulational instability of the nonlinear Schro\ifmmode\ddot\else\textasciidieresis\fi{}dinger equation. We also consider a nonlinear regime slightly above the instability threshold. The results obtained can be applied to the problem of single-particle tunneling through an ultrashort junction in the presence of multiparticle interaction. Our prediction is that the instability gives rise to an ac component in the transmitted current.

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