Global Attractor for 1D Dirac Field Coupled to Nonlinear Oscillator

Abstract

Global attraction to solitary waves is proved for a model mathbf {U}(1)-invariant nonlinear 1D Dirac equation coupled to a nonlinear oscillator: each finite energy solution converges as trightarrow pm infty to a set of all “nonlinear eigenfunctions” of the form psi _1(x)e^{-iomega _1 t}+psi _2(x)e^{-iomega _2 t}. The global attraction is caused by nonlinear energy transfer from lower harmonics to continuous spectrum and subsequent dispersive radiation. We justify this mechanism by a strategy based on inflation of spectrum by the nonlinearity. We show that any omega-limit trajectory has the time-spectrum in the spectral gap [-m,m] and satisfies the original equation.Then the application of the Titchmarsh convolution theorem reduces the spectrum of the omega-limit trajectory to two harmonics omega _jin [-m,m], j =1,2.