Weak attractorof the Klein-Gordon field in discrete space-timeinteracting with a nonlinear oscillator

Andrew Comech

doi:10.3934/dcds.2013.33.2711

Article Contents

2013, Volume 33, Issue 7: 2711-2755. Doi: 10.3934/dcds.2013.33.2711

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

Andrew Comech^1,

1.
Texas A&M University, College Station, Texas 77843

Received: March 31, 2012

Revised: September 30, 2012

Published: June 2013

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Abstract

We consider the $\mathbf{U}(1)$-invariant nonlinear Klein-Gordon equation in discrete space and discrete time, which is the discretization of the nonlinear continuous Klein-Gordon equation. To obtain this equation, we use the energy-conserving finite-difference scheme of Strauss-Vazquez. We prove that each finite energy solution converges as $T → ± ∞$ to the finite-dimensional set of all multifrequency solitary wave solutions with one, two, and four frequencies. The components of the solitary manifold corresponding to the solitary waves of the first two types are generically two-dimensional, while the component corresponding to the last type is generically four-dimensional. The attraction to the set of solitary waves is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent radiation. For the proof, we develop the well-posedness for the nonlinear wave equation in discrete space-time, apply the technique of quasimeasures, and also obtain the version of the Titchmarsh convolution theorem for distributions on the circle.

Keywords:

Weak attractors for dispersive equations,
finite-difference schemes,
long-time asymptotics.

Mathematics Subject Classification: 35B40, 35B41, 35C08, 37K40, 37L30, 65M06, 65M12, 65N06, 81Q05.

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