Variational approach to multimode nonlinear optical fibers

We analyze the spatiotemporal solitary waves of a graded-index multimode optical fiber with a parabolic transverse index profile. Using the nonpolynomial Schr\"odinger equation approach, we derive an effective one-dimensional Lagrangian associated with the Laguerre-Gauss modes with a generic radial mode number p and azimuthal index m. We show that the form of the equations of motion for any Laguerre-Gauss mode is particularly simple, and we derive the critical power for the collapse for every mode. By solving the nonpolynomial Schr\"odinger equation, we provide a comparison of the stationary mode profiles in the radial and temporal coordinates.