On the degenerate soliton solutions of the focusing nonlinear Schrödinger equation

We characterize the N-soliton solutions of the focusing nonlinear Schrödinger (NLS) equation with degenerate velocities, i.e., solutions in which two or more soliton velocities are the same, which are obtained when two or more discrete eigenvalues of the scattering problem have the same real parts. We do so by employing the operator formalism developed by one of the authors to express the N-soliton solution of the NLS equation in a convenient form. First we analyze soliton solutions with fully degenerate velocities (a so-called multi-soliton group), clarifying their dependence on the soliton parameters. We then consider the dynamics of soliton groups interaction in a general N-soliton solution. We compute the long-time asymptotics of the solution and we quantify the interaction-induced position and phase shifts of each non-degenerate soliton as well as the interaction-induced changes in the center of mass and soliton parameters of each soliton group.