536 research outputs found
Generation of oblique dark solitons in supersonic flow of Bose-Einstein condensate past an obstacle
Nonlinear and dispersive properties of Bose-Einstein condensate (BEC) provide a possibility of formation of various nonlinear structures such as vortices and bright and dark
solitons (see, e.g., [1]). Yet another type of nonlinear wave patterns has been observed in
a series of experiments on the BEC flow past macroscopic obstacles [2]. In [3] these structures have been associated with spatial dispersive shock waves. Spatial dispersive shock
waves represent dispersive analogs of the the well-known viscous spatial shocks (oblique
jumps of compression) occurring in supersonic flows of compressible fluids past obstacles.
In a viscous fluid, the shock can be represented as a narrow region within which strong
dissipation processes take place and the thermodynamic parameters of the flow undergo
sharp change. On the contrary, if viscosity is negligibly small compared with dispersion
effects, the shock discontinuity resolves into an expanding in space oscillatory structure
which transforms gradually, as the distance from the obstacle increases, into a \fan" of
stationary solitons. If the obstacle is small enough, then such a \fan" reduces to a single
spatial dark soliton [4]. Here we shall present the theory of these new structures in BEC
Interactions of large amplitude solitary waves in viscous fluid conduits
The free interface separating an exterior, viscous fluid from an intrusive
conduit of buoyant, less viscous fluid is known to support strongly nonlinear
solitary waves due to a balance between viscosity-induced dispersion and
buoyancy-induced nonlinearity. The overtaking, pairwise interaction of weakly
nonlinear solitary waves has been classified theoretically for the Korteweg-de
Vries equation and experimentally in the context of shallow water waves, but a
theoretical and experimental classification of strongly nonlinear solitary wave
interactions is lacking. The interactions of large amplitude solitary waves in
viscous fluid conduits, a model physical system for the study of
one-dimensional, truly dissipationless, dispersive nonlinear waves, are
classified. Using a combined numerical and experimental approach, three classes
of nonlinear interaction behavior are identified: purely bimodal, purely
unimodal, and a mixed type. The magnitude of the dispersive radiation due to
solitary wave interactions is quantified numerically and observed to be beyond
the sensitivity of our experiments, suggesting that conduit solitary waves
behave as "physical solitons." Experimental data are shown to be in excellent
agreement with numerical simulations of the reduced model. Experimental movies
are available with the online version of the paper.Comment: 13 pages, 4 figure
Unified Approach to KdV Modulations
We develop a unified approach to integrating the Whitham modulation
equations. Our approach is based on the formulation of the initial value
problem for the zero dispersion KdV as the steepest descent for the scalar
Riemann-Hilbert problem, developed by Deift, Venakides, and Zhou, 1997, and on
the method of generating differentials for the KdV-Whitham hierarchy proposed
by El, 1996. By assuming the hyperbolicity of the zero-dispersion limit for the
KdV with general initial data, we bypass the inverse scattering transform and
produce the symmetric system of algebraic equations describing motion of the
modulation parameters plus the system of inequalities determining the number
the oscillating phases at any fixed point on the - plane. The resulting
system effectively solves the zero dispersion KdV with an arbitrary initial
data.Comment: 27 pages, Latex, 5 Postscript figures, to be submitted to Comm. Pure.
Appl. Mat
Two-dimensional periodic waves in supersonic flow of a BoseāEinstein condensate
Stationary periodic solutions of the two-dimensional GrossāPitaevskii equation are obtained and analysed for different parameter values in the context of the problem of a supersonic flow of a BoseāEinstein condensate past an obstacle. The asymptotic connections with the corresponding periodic solutions of the Kortewegāde Vries and nonlinear Schrƶdinger equations are studied and typical spatial wave distributions are discussed
Dam break problem for the focusing nonlinear Schr\"odinger equation and the generation of rogue waves
We propose a novel, analytically tractable, scenario of the rogue wave
formation in the framework of the small-dispersion focusing nonlinear
Schr\"odinger (NLS) equation with the initial condition in the form of a
rectangular barrier (a "box"). We use the Whitham modulation theory combined
with the nonlinear steepest descent for the semi-classical inverse scattering
transform, to describe the evolution and interaction of two counter-propagating
nonlinear wave trains --- the dispersive dam break flows --- generated in the
NLS box problem. We show that the interaction dynamics results in the emergence
of modulated large-amplitude quasi-periodic breather lattices whose amplitude
profiles are closely approximated by the Akhmediev and Peregrine breathers
within certain space-time domain. Our semi-classical analytical results are
shown to be in excellent agreement with the results of direct numerical
simulations of the small-dispersion focusing NLS equation.Comment: 29 pages, 15 figures, major revisio
Evolution of solitary waves and undular bores in shallow-water flows over a gradual slope with bottom friction
This paper considers the propagation of shallow-water solitary and nonlinear periodic waves over a gradual slope with bottom friction in the framework of a variable-coefficient Korteweg-de Vries equation. We use the Whitham averaging method, using a recent development of this theory for perturbed integrable equations. This general approach enables us not only to improve known results on the adiabatic evolution of isolated solitary waves and periodic wave trains in the presence of variable topography and bottom friction, modelled by the Chezy law, but also, importantly, to study the effects of these factors on the propagation of undular bores, which are essentially unsteady in the system under consideration. In particular, it is shown that the combined action of variable topography and bottom friction generally imposes certain global restrictions on the undular bore propagation so that the evolution of the leading solitary wave can be substantially different from that of an isolated solitary wave with the same initial amplitude. This non-local effect is due to nonlinear wave interactions within the undular bore and can lead to an additional solitary wave amplitude growth, which cannot be predicted in the framework of the traditional adiabatic approach to the propagation of solitary waves in slowly varying media
Wave Breaking and the Generation of Undular Bores in an Integrable Shallow Water System
The generation of an undular bore in the vicinity of a waveābreaking point is considered for the integrable KaupāBoussinesq (KB) shallow water system. In the framework of the Whitham modulation theory, an analytic solution of the GurevichāPitaevskii type of problem for a generic ācubicā breaking regime is obtained using a generalized hodograph transform, and a further reduction to a linear EulerāPoisson equation. The motion of the undular bore edges is investigated in detail
Solitonic dispersive hydrodynamics: theory and observation
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.Comment: 8 pages, 5 figure
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