Asymptotic stability of $N$-solitons in the cubic NLS equation

In this article we consider the Cauchy problem for the cubic focusing nonlinear Schr"o-dinger (NLS) equation on the line with initial datum close to a particular $N$-soliton. Using inverse scattering and the $\bar{\mathrm{\partial }}$ method we establish the decay of the $L^{\mathrm{\infty }}$ norm of the residual term in time.