Weak attractor of the Klein-Gordon field in discrete space-time interacting with a nonlinear oscillator

Abstract

We consider the $\mathbf{U}(1)$-invariantnonlinear Klein-Gordon equationin discrete space and discrete time,which is the discretizationof the nonlinear continuous Klein-Gordon equation.To obtain this equation, we usethe energy-conserving finite-difference scheme of Strauss-Vazquez.We prove thateach finite energy solution converges as $T → ± ∞$to the finite-dimensionalset of all multifrequency solitary wave solutionswith one, two, and four frequencies.The components of the solitary manifoldcorresponding to the solitary wavesof the first two types are generically two-dimensional,while thecomponent corresponding to the last typeis generically four-dimensional.The attractionto the set of solitary wavesis caused by the nonlinear energy transferfrom lower harmonics to the continuous spectrumand subsequent radiation.For the proof, we developthe well-posedness for the nonlinear wave equationin discrete space-time,apply the technique of quasimeasures,and also obtain the version of the Titchmarsh convolution theoremfor distributions on the circle.