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Hamiltonian-versus-energy diagrams in soliton theory Nail Akhmediev and Adrian Ankiewicz

Summary: Hamiltonian-versus-energy diagrams in soliton theory
Nail Akhmediev and Adrian Ankiewicz
Australian Photonics CRC, Optical Sciences Centre, The Australian National University, Canberra 0200,
Australian Capital Territory, Australia
Roger Grimshaw
Department of Mathematics and Statistics, Monash University, Clayton, Victoria 3168, Australia
Received 10 September 1998; revised manuscript received 12 January 1999
Parametric curves featuring Hamiltonian versus energy are useful in the theory of solitons in conservative
nonintegrable systems with local nonlinearities. These curves can be constructed in various ways. We show
here that it is possible to find the Hamiltonian H and energy Q for solitons of non-Kerr-law media with local
nonlinearities without specific knowledge of the functional form of the soliton itself. More importantly, we
show that the stability criterion for solitons can be formulated in terms of H and Q only. This allows us to
derive all the essential properties of solitons based only on the concavity of the curve H vs Q. We give
examples of these curves for various nonlinearity laws and show that they confirm the general principle. We
also show that solitons of an unstable branch can transform into solitons of a stable branch by emitting small
amplitude waves. As a result, we show that simple dynamics like the transformation of a soliton of an unstable
branch into a soliton of a stable branch can also be predicted from the H-Q diagram.
S1063-651X 99 09805-0
PACS number s : 42.65. k, 47.20.Ky, 47.27.Te


Source: Akhmediev, Nail - Research School of Physical Sciences and Engineering, Australian National University


Collections: Physics; Engineering