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

Many physical, chemical, and biological systems are described by partial differential equations which support localized soliton solutions. Usually, these cannot be obtained analytically and so we try to use a '' trial function '' which gives a good approximation to the pulse solution. This can provide direct insight into the nature of the system's behavior for a wide range of input parameters. The Lagrangian approach or the '' method of moments '' can be used with the trial function for this purpose. The two techniques have been developed independently but never directly compared. A question arises: Do the two approaches result in the same dynamical system of coupled ordinary differential equations? This work provides at least a partial answer to this question. For a realistic and common form of the phase dependence, we find that the same resultant system is obtained in each case. For solitons which move, or have a more complicated phase dependence, the resulting systems are different.