Internal dynamics of a vector soliton in a nonlinear optical fiber

We analyze the dynamics of a vector soliton governed by a nearly integrable system of coupled nonlinear Schr\"odinger equations. Inserting a Gaussian ansatz into the Lagrangian density, we derive a system of ordinary differential equations for the evolution of the ansatz parameters. We find a continuous family of stationary solutions to these equations which can be interpreted as vector solitons with an arbitrary polarization. Examining small internal vibrations of the vector soliton, we find three eigenmodes, of which only two were previously known. The additional internal oscillation eigenmode gives rise to antisymmetric oscillations of the symmetric soliton (45\ifmmode^\circ\else\textdegree\fi{} polarization). We also find the small-vibration eigenmodes for arbitrary polarization, though in an implicit form. Additionally, we find a threshold value of the relative velocity of the two polarizations that leads to splitting of the vector soliton for arbitrary polarization.