On the nonlinear Schrödinger equation with a boundary condition involving a time derivative of the field

We study the nonlinear Schrödinger equation on the half-line with a new boundary condition presented by Zambon. This new boundary involves a time derivative of the field and was already shown by Zambon to be integrable. In this paper we re-establish the integrability of such a boundary both by using the Sklyanin’s formalism and by using the tool of Bäcklund transformations. Moreover, we present a method to derive explicit formulae for multi-soliton solutions of the boundary problem by virtue of the Darboux transformation method in conjunction with a boundary dressing technique.