Lagrangian approach for dark solitons

Abstract

We propose a variational formalism for dark solitons based on a nontrivial modification of the standard Lagrangian of the nonlinear Schrödinger equation. Our approach allows to consider a variety of asymptotically nonvanishing localised solutions for applying a variational technique together with the correct definition of momentum and energy. We demonstrate, on several particular examples, how this variational approach works, and we also point out its suitability for more general problems, e.g. those involving the analysis of the dark soliton interaction as well as the perturbation-induced dynamics of dark solitons and optical vortex solitons.