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Global attraction to solitary waves for Klein-Gordon equation with mean field interaction. (English) Zbl 1177.35201

Summary: We consider a \(\mathbf U(1)\)-invariant nonlinear Klein-Gordon equation in dimension \(n\geq1\), self-interacting via the mean field mechanism. We analyze the long-time asymptotics of finite energy solutions and prove that, under certain generic assumptions, each solution converges as \(t\rightarrow \pm \infty \) to the two-dimensional set of all “nonlinear eigenfunctions” of the form \(\varphi (x)e ^{- i\omega t}\). This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q51 Soliton equations
35B40 Asymptotic behavior of solutions to PDEs
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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