<p>This paper is addressed to study asymptotic behavior for the Korteweg-de Vries equation with rapid oscillations, namely, the Korteweg-de Vries equation with rapidly oscillating potential and rapidly oscillating boundary force. In order to describe its asymptotic behavior, we establish the averaging principle for this system, this is important from both physical and mathematical standpoints. More precisely, two kinds of averaging principle is established, one is Bogoliubov first averaging principle for the Korteweg-de Vries equation on a finite time interval, the other is the Bogoliubov second averaging principle for the Korteweg-de Vries equation on the entire axis.</p>

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Asymptotic Behavior for the Korteweg-de Vries Equation with Rapid Oscillations

  • Mo Chen

摘要

This paper is addressed to study asymptotic behavior for the Korteweg-de Vries equation with rapid oscillations, namely, the Korteweg-de Vries equation with rapidly oscillating potential and rapidly oscillating boundary force. In order to describe its asymptotic behavior, we establish the averaging principle for this system, this is important from both physical and mathematical standpoints. More precisely, two kinds of averaging principle is established, one is Bogoliubov first averaging principle for the Korteweg-de Vries equation on a finite time interval, the other is the Bogoliubov second averaging principle for the Korteweg-de Vries equation on the entire axis.