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.


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
Full Text: DOI arXiv


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