Normal form approach to near-linear dynamics of modified KdV equation

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. (C) 2018 Elsevier Inc. All rights reserved.