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Rigorous description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations

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  • It is the purpose of this paper to prove error estimates for the approximate description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations, like the Korteweg--de Vries (KdV) or the Nonlinear Schrödinger (NLS) equation. The proofs are based on a discrete Bloch wave transform of the underlying infinite-dimensional system of coupled ODEs. After this transform the existing proof for the associated approximation theorem for the NLS approximation used for the approximate description of oscillating wave packets in dispersive PDE systems transfers almost line for line. In contrast, the proof of the approximation theorem for the KdV approximation of long waves is less obvious. In a special situation we prove a first approximation result.
    Mathematics Subject Classification: Primary: 34E20; Secondary: 35Q53, 35Q55.


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