Comparison of Lagrangian approach and method of moments for reducing dimensionality of soliton dynamical systems

For equations that cannot be solved exactly, the trial function approach to modelling soliton solutions represents a useful approximate technique. It has to be supplemented with the Lagrangian technique or the method of moments to obtain a finite dimensional dynamical system which can be analyzed more easily than the original partial differential equation. We compare these two approaches. Using the cubic-quintic complex Ginzburg-Landau equation as an example, we show that, for a wide class of plausible trial functions, the same system of equations will be obtained. We also explain where the two methods differ.