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Register a new userNumber of solitons produced from a large initial pulse in the generalized NLS dispersive hydrodynamics theory
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We show that the number of solitons produced from an arbitrary initial pulse of the simple wave type can be calculated analytically if its evolution is governed by a generalized nonlinear Schr{o}dinger equation provided this number is large enough. The final result generalizes the asymptotic formula derived for completely integrable nonlinear wave equations like the standard NLS equation with the use of the inverse scattering transform method.
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The theory of motion of edges of dispersive shock waves generated after wave breaking of simple waves is developed. It is shown that this motion obeys Hamiltonian mechanics complemented by a Hopf-like equation for evolution of the background flow that interacts with edge wave packets or edge solitons. A conjecture about existence of a certain symmetry between equations for the small-amplitude and soliton edges is formulated. In case of localized simple wave pulses propagating through a quiescent medium this theory provided a new approach to derivation of an asymptotic formula for the number of solitons produced eventually from such a pulse.
Ubiquitous nonlinear waves in dispersive media include localized solitons and extended hydrodynamic states such as dispersive shock waves. Despite their physical prominence and the development of thorough theoretical and experimental investigations of each separately, experiments and a unified theory of solitons and dispersive hydrodynamics are lacking. Here, a general soliton-mean field theory is introduced and used to describe the propagation of solitons in macroscopic hydrodynamic flows. Two universal adiabatic invariants of motion are identified that predict trapping or transmission of solitons by hydrodynamic states. The result of solitons incident upon smooth expansion waves or compressive, rapidly oscillating dispersive shock waves is the same, an effect termed hydrodynamic reciprocity. Experiments on viscous fluid conduits quantitatively confirm the soliton-mean field theory with broader implications for nonlinear optics, superfluids, geophysical fluids, and other dispersive hydrodynamic media.
We report the experimental observation of multiple dispersive waves emitted in the anomalous dispersion region of an optical fiber from a train of dark solitons. Each individual dispersive wave can be associated to one particular dark soliton of the train, using phase-matching arguments involving higher-order dispersion and soliton velocity. For a large number of dark solitons (>10), we observe the formation of a continuum associated with the efficient emission of dispersive waves.
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The problem of stability and spectrum of linear excitations of a soliton (kink) of the dispersive sine-Gordon and $varphi^4$ - equations is solved exactly. It is shown that the total spectrum consists of a discrete set of frequencies of internal modes and a single band spectrum of continuum waves. It is indicated by numerical simulations that a translation motion of a single soliton in the highly dispersive systems is accompanied by the arising of its internal dynamics and, in some cases, creation of breathers, and always by generation of the backward radiation. It is shown numerically that a fast motion of two topological solitons leads to a formation of the bound soliton complex in the dispersive sine-Gordon system.
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