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 $$t\rightarrow \pm \infty $$ to a 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 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 $$\omega _j\in [-m,m]$$, $$j =1,2$$.

Locations

• Communications in Mathematical Physics - View - PDF
• arXiv (Cornell University) - View - PDF
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