Abstract We analyze the spatiotemporal solitary waves of a graded-index multimode optical fiber with a parabolic transverse index profile. Using the nonpolynomial Schrödinger 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 Schrödinger equation, we provide a comparison of the stationary mode profiles in the radial and temporal coordinates.