Global solutions of quasi-linear Hamiltonian mKdV equation

Abstract

We study the initial value problem of quasi-linear Hamiltonian mKdV equations. Our goal is to prove the global-in-time existence of a solution given sufficiently smooth, localized, and small initial data. To achieve this, we utilize the bootstrap argument, Sobolev energy estimates, and the dispersive estimate. This proof relies on the space–time resonance method, as well as a bilinear estimate developed by Germain, Pusateri, and Rousset.