Multisolitons for the cubic NLS in 1-d and their stability

For both the cubic Nonlinear Schrödinger Equation (NLS) as well as the modified Korteweg-de Vries (mKdV) equation in one space dimension we consider the set ${\mathbf{M}}_N$ $\mathbf {M}_{N}$ of pure $N$ $N$ -soliton states, and their associated multisoliton solutions. We prove that (i) the set ${\mathbf{M}}_N$ $\mathbf {M}_{N}$ is a uniformly smooth manifold, and (ii) the ${\mathbf{M}}_N$ $\mathbf {M}_{N}$ states are uniformly stable in $H^s$ $H^{s}$ , for each $s>-\frac{1}{2}$ $s>-\frac{1}{2}$ .

One main tool in our analysis is an iterated Bäcklund transform, which allows us to nonlinearly add a multisoliton to an existing soliton free state (the soliton addition map) or alternatively to remove a multisoliton from a multisoliton state (the soliton removal map). The properties and the regularity of these maps are extensively studied.