Global attractor for 1D Dirac field coupled to nonlinear oscillator

The long-time asymptotics is analyzed for all finite energy solutions to a model $mathbf{U}(1)$-invariant nonlinear Dirac equation in one dimension, coupled to a nonlinear oscillator: {it each finite energy solution} converges as $ttopminfty$ to the set of all `nonlinear eigenfunctions of the form $(psi_1(x)e^{-iomega_1 t},psi_2(x)e^{-iomega_2 t})$. The {it global attraction} is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation. We justify this mechanism by the strategy based on emph{inflation of spectrum by the nonlinearity}. We show that any {it omega-limit trajectory} has the time-spectrum in the spectral gap $[-m,m]$ and satisfies the original equation. This equation implies the key {it spectral inclusion} for spectrum of the nonlinear term. Then the application of the Titchmarsh convolution theorem reduces the spectrum of $j$-th component of the omega-limit trajectory to a single harmonic $omega_jin[-m,m]$, $j=1,2$.