Long time asymptotics of a perturbed modified KdV equation

We derive full asymptotics of the modified KdV equation (mKdV) with a higher-order perturbative term. We make use of the perturbative theory of infinite-dimensional integrable systems developed by P. Deift and X. Zhou \cite{DZ-2}, and some new and simpler proofs of certain $L^{\mathrm{\infty }}$ bounds and $L^p$ a priori estimates developed recently in \cite{CLT}. We show that the perturbed equation exhibits the same long-time behavior as the completely integrable mKdV.