Abstract: 
We consider the quasilinear Hamiltonian mKdV equation $\varphi_t+\partial_x^3\varphi+\partial_x (\varphi^3)+\partial_x[c(\varphi)\partial_x(c(\varphi)\partial_x\varphi)]=0$. We prove that when the initial data is sufficiently smooth, localized and small, the solution exists globalintime. 
