Abstract
In this paper, we implement normal form reduction to the periodic modified Korteweg–de Vries (mKdV) equation to investigate the behavior of a solution when a subtle high-frequency initial data is given. We use differentiation by parts to decompose the equation into resonant and non-resonant parts and provide some nonlinear estimates for each term. If a subtle high-frequency initial data is given, a solution of the mKdV equation can be approximated by a solution of the linearized mKdV equation for large times.
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