Variationalism and Empirio-Criticism. (Exact and Variational Approaches to Fibre Optics Equations)

Abstract

There were many interesting attempts to describe solitary wave solutions of nonlinear partial differential equations as particles with internal degrees of freedom and satisfying an effective finite dimensional Lagrangian equations (see, for example [1], [2]). A simplified “Variational Approach” to Lagrangian nonlinear partial differential equations, formulated by D.Anderson in [3], becomes very popular, specially in the non-linear optics community, and hundreds of papers published every year are actually rely on it. Even during the current school quite a big proportion of theoretical works is based upon the variational approach. It is very easy to understand why this approach is so attractive:

• it is a very universal method, suitable for equations in any dimensions, with external forces and potentials (it does not matter at all whether the equation is integrable or not),

• often it gives results which look quite similar to numeric simulations,

• it is an extremely simple approach — one can learn it in a half an hour, obtain a result in the next few days and in a week write a reasonably looking paper which may be well accepted by many very respectable refereed journals, including the Physical Review and specialised optical journals,

• a simplified finite dimensional dynamical system obtained via this approach may have interesting and rich properties. It is much easier to study such a reduced system than the original problem, etc.