Hamiltonian-versus-energy diagrams in soliton theory

Parametric curves featuring Hamiltonian versus energy are useful in the the ory of solitons in conservative nonintegrable systems with local nonlineari ties. 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 s tability criterion for solitons can be formulated in terms of PI and Q only . This allows us to derive all the essential properties of solitons based o nly 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 princi ple. We also show that solitons of an unstable branch can transform into so litons of a stable branch by emitting small amplitude waves. Asa result, we show that simple dynamics Like the transformation of a soliton of an unsta ble branch into a soliton of a stable branch can also be predicted from the H-Q diagram. [S1063-651X(99)09805-0].