  We investigate the spatial beam dynamics inside a multimode graded-index
fiber under Raman amplification by adopting a semi-analytical variational
approach. The variational analysis provides us with four coupled ordinary
differential equations that govern the beam's dynamics under Raman gain and are
much faster to solve numerically compared to the full nonlinear wave equation.
Their solution also provides considerable physical insight and allows us to
study the impact of important nonlinear phenomena such as self-focusing and
cross-phase modulation. We first show that the variational results corroborate
well with full numerical simulations and then use them to investigate the
signal's dynamics under different initial conditions such as the initial widths
of the pump and signal beams. This allows us to quantify the conditions under
which the quality of a signal beam can improve, without collapse of the beam
owing to self-focusing. While time-consuming full simulations may be needed
when gain saturation and pump depletion must be included, the variational
method is useful for gaining valuable physical insight and for studying
dependence of the amplified beam's width and amplitude on various physical
parameters in a faster fashion.
