We consider scattering of quantum particles by a potential which includes a delta-function whose amplitude contains a term proportional to the density of particles. This is the simplest model of the single-particle tunneling through an ultrashort junction in the presence of the multiparticle interactions. Solution of the scattering problem in this nonlinear model is non-unique 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 has a threshold in the density of particles as well as in the single-particle energy. The instability must manifest itself, generating an ac component in the transmitted current.