Summary: Comparison of Lagrangian approach and method of moments
for reducing dimensionality of soliton dynamical systems
Adrian Ankiewicz and Nail Akhmediev
Optical Sciences Group, Research School of Physical Sciences and Engineering,
The Australian National University, Canberra, Australian Capital Territory 0200, Australia
Received 9 May 2008; accepted 8 August 2008; published online 9 September 2008
For equations that cannot be solved exactly, the trial function approach to modelling soliton solu-
tions 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 ap-
proaches. Using the cubic-quintic complex GinzburgLandau 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. © 2008 American Institute of Physics.
Many physical, chemical, and biological systems are de-
scribed by partial differential equations which support
localized soliton solutions. Usually, these cannot be ob-
tained 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 sys-