  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.
