Multisoliton solutions of the vector nonlinear Schr\"odinger equation (Kulish-Sklyanin model) and the vector mKdV equation

There exist two natural vector generalizations of the completely integrable nonlinear Schr\"odinger (NLS) equation in $1+1$ dimensions: the well-known Manakov model and the lesser-known Kulish-Sklyanin model. In this paper, we propose a binary Darboux (or Zakharov-Shabat dressing) transformation that can be directly applied to the Kulish-Sklyanin model. By deriving a simple closed expression for iterations of the binary Darboux transformation, we obtain an explicit formula for the $N$-soliton solution of the Kulish-Sklyanin model under vanishing boundary conditions. Because the third-order symmetry of the vector NLS equation can be reduced to a vector generalization of the modified KdV (mKdV) equation, we can also obtain multisoliton (or multi-breather) solutions of the vector mKdV equation in closed form.