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On the global attraction to solitary waves for the Klein-Gordon equation coupled to a nonlinear oscillator. (English) Zbl 1096.35020

Summary: The long-time asymptotics are analyzed for all finite energy solutions to a model \(U(1)\)-invariant nonlinear Klein-Gordon equation in one dimension, with the nonlinearity concentrated at a point. Our main result is that each finite energy solution converges as \(t\rightarrow \pm \infty\) to the set of ‘nonlinear eigenfunctions’ \(\psi (x)e^{-\mathrm i\omega t}\).

MSC:

35B41 Attractors
35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35Q51 Soliton equations
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
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