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

Abstract

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 \(\mathbf {M}_{N}\) of pure \(N\)-soliton states, and their associated multisoliton solutions. We prove that (i) the set \(\mathbf {M}_{N}\) is a uniformly smooth manifold, and (ii) the \(\mathbf {M}_{N}\) states are uniformly stable in \(H^{s}\), for each \(s>-\frac{1}{2}\).

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.