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Global attractor for a nonlinear oscillator coupled to the Klein-Gordon field. (English) Zbl 1131.35003

The authors study long-time asymptotic behaviour for all finite energy solutions to Cauchy problem for the model of the one-dimensional \(U(1)\)-invariant \((F(e^{ - i\theta }\psi ) = e^{ - i\theta }F(\psi)\), \(\psi \in \mathbb{C}\), \(\theta \in \mathbb{R})\) nonlinear Klein-Gordon equation with nonlinearity concentrated at a single point
\[ \ddot {\psi }(x,t) = {\psi }''(x,t) - m^2\psi (x,t) + \delta (x)F(\psi (0,t)),\quad \psi (x,0) = \psi _0 (x),\quad \dot {\psi }(x,0) = \pi _0 (x): \]
each finite energy solution converges for \(t \to \pm \infty \) to the set of all nonlinear eigenfunctions of the form \(\psi(x)\exp (-i\omega t)\). The global attractor for all finite energy solutions is considered. In particular it is proved that for a U(1)-invariant dispersive Hamiltonian system the global attractor is finite-dimensional and is formed by solitary waves which are finite energy solutions of the type \(\psi_\omega(x,t)=\psi_\omega(x)e^{i\omega t}\), \(\omega\in\mathbb{C}\). It is shown that the global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.

MSC:

35B41 Attractors
35Q40 PDEs in connection with quantum mechanics
81T10 Model quantum field theories
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35B40 Asymptotic behavior of solutions to PDEs
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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