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 $t\to\pm\infty$ to the set of all `nonlinear eigenfunctions' of the form $(\psi_1(x)e^{-i\omega_1 t},\psi_2(x)e^{-i\omega_2 t})$. 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 $\omega_j\in[-m,m]$, $j=1,2$.