Hamiltonian-versus-energy diagrams in soliton theory

Nail Akhmediev, Adrian Ankiewicz, Roger Grimshaw
1999 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics  
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 ͑H͒ and energy ͑Q͒ 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
more » ... ns 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.
doi:10.1103/physreve.59.6088 pmid:11969594 fatcat:z7qte6vxzfdcbfmopkxsbyvomu