 
Summary: Stability analysis for solitons in planar waveguides,
fibres and couplers using Hamiltonian concepts
A. Ankiewicz and N. Akhmediev
Abstract: The use of Hamiltonianversusenergy (HVE) curves for localised optical soliton
solutions is a powerful method for studying systems with an infinite number of degrees of freedom.
These curves are useful for analysing the range of existence and stability of solitons. Detailed
analysis of HVE curves and their special points are given. The authors illustrate their conclusions
with several new examples which show the usefulness of the concept. The main example is related
to nonKerrtype solitons in planar waveguides, although examples of solitons in fibres and
couplers are also provided. Specifically it is shown that, in the special case of a twodimensional
beam in a Kerr medium, the curve contracts to a point. It is also demonstrated that, in some cases, it
is possible to find the Hamiltonian and energy without knowledge of the functional form of the
soliton itself. The authors explain how this can be used in various aspects of soliton theory.
1 Introduction
The concept of the Hamiltonian is fundamental in
mechanics [1] and more generally in the study of
conservative dynamical systems with a finite [2] or infinite
[3] number of degrees of freedom. It has turned out to be
especially useful in the soliton theory of completely
integrable systems [3]. The theory of systems which are
