Two-dimensional periodic waves in supersonic flow of a Bose–Einstein condensate

El, Gennady; Gladush, Yuriy G.; Камчатнов, А. М.

doi:10.1088/1751-8113/40/4/003

Cited by 17 publications

(20 citation statements)

References 16 publications

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“…This does not contradict the orientation of small amplitude dispersive waves because their propagation direction is orthogonal to the direction of constant phase, hence is outside the Mach cone. These restrictions on wave orientation have been previously described in the context of supersonic NLS flow past a small impurity where oblique solitary waves and dispersive waves are generated [9]. A calculation shows that the a → 0 limit in eq.…”

Section: Properties Of the Stationary Nls Equationsupporting

confidence: 51%

Oblique Spatial Dispersive Shock Waves in Nonlinear Schrödinger Flows

Hoefer

El²,

Камчатнов³

2017

SIAM J. Appl. Math.

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Abstract. In dispersive media, hydrodynamic singularities are resolved by coherent wavetrains known as dispersive shock waves (DSWs). Only dynamically expanding, temporal DSWs are possible in one-dimensional media. The additional degree of freedom inherent in two-dimensional media allows for the generation of time-independent DSWs that exhibit spatial expansion. Spatial oblique DSWs, dispersive analogs of oblique shocks in classical media, are constructed utilizing Whitham modulation theory for a class of nonlinear Schrödinger boundary value problems. Self-similar, simple wave solutions of the modulation equations yield relations between the DSW's orientation and the upstream/downstream flow fields. Time dependent numerical simulations demonstrate a convective or absolute instability of oblique DSWs in supersonic flow over obstacles. The convective instability results in an effective stabilization of the DSW.

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Section: Properties Of the Stationary Nls Equationsupporting

confidence: 51%

Oblique Spatial Dispersive Shock Waves in Nonlinear Schrödinger Flows

Hoefer

El²,

Камчатнов³

2017

SIAM J. Appl. Math.

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“…The system (9.1) has a family of stationary, periodic solutions [251] and is amenable, in principle, to a general modulation analysis using the DSW fitting method. Care must be taken due to the presence of a 2D convective instability [252,253], a feature, not occuring in 1D flows.…”

Section: Supersonic Dispersive Flows Past Obstacles and 2d Oblique Dswsmentioning

confidence: 99%

Dispersive shock waves and modulation theory

Hoefer

2016

Physica D: Nonlinear Phenomena

294

590

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There is growing physical and mathematical interest in the hydrodynamics of dissipationless/dispersive media. Since G.~B.~Whitham's seminal publication fifty years ago that ushered in the mathematical study of dispersive hydrodynamics, there has been a significant body of work in this area. However, there has been no comprehensive survey of the field of dispersive hydrodynamics. Utilizing Whitham's averaging theory as the primary mathematical tool, we review the rich mathematical developments over the past fifty years with an emphasis on physical applications. The fundamental, large scale, coherent excitation in dispersive hydrodynamic systems is an expanding, oscillatory dispersive shock wave or DSW. Both the macroscopic and microscopic properties of DSWs are analyzed in detail within the context of the universal, integrable, and foundational models for uni-directional (Korteweg-de Vries equation) and bi-directional (Nonlinear Schr\"{o}dinger equation) dispersive hydrodynamics. A DSW fitting procedure that does not rely upon integrable structure yet reveals important macroscopic DSW properties is described. DSW theory is then applied to a number of physical applications: superfluids, nonlinear optics, geophysics, and fluid dynamics. Finally, we survey some of the more recent developments including non-classical DSWs, DSW interactions, DSWs in perturbed and inhomogeneous environments, and two-dimensional, oblique DSWs.Comment: review article, 68 pages, 52 figure

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“…Remarkably, the distribution (139) is invariant with respect to the evolution, up to a breaking point at T = T b , of the Riemann invariant λ + , described by the simple-wave equation (50) (which is consistent with the dispersionless limit of the NLS equation (12)). Indeed, it is not difficult to show that (50) implies that ∂ ∂T (λ − λ + )(λ + 1) dY = 0 .…”

Section: Trailing Edgementioning

confidence: 99%

Two-dimensional supersonic nonlinear Schrödinger flow past an extended obstacle

Камчатнов

Khodorovskii

et al. 2009

Phys. Rev. E

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Supersonic flow of a superfluid past a slender impenetrable macroscopic obstacle is studied in the framework of the two-dimensional defocusing nonlinear Schrödinger (NLS) equation. This problem is of fundamental importance as a dispersive analogue of the corresponding classical gas-dynamics problem. Assuming the oncoming flow speed sufficiently high, we asymptotically reduce the original boundary-value problem for a steady flow past a slender body to the one-dimensional dispersive piston problem described by the nonstationary NLS equation, in which the role of time is played by the stretched x-coordinate and the piston motion curve is defined by the spatial body profile. Two steady oblique spatial dispersive shock waves (DSWs) spreading from the pointed ends of the body are generated in both half-planes. These are described analytically by constructing appropriate exact solutions of the Whitham modulation equations for the front DSW and by using a generalized Bohr-Sommerfeld quantization rule for the oblique dark soliton fan in the rear DSW. We propose an extension of the traditional modulation description of DSWs to include the linear "ship wave" pattern forming outside the nonlinear modulation region of the front DSW. Our analytic results are supported by direct 2D unsteady numerical simulations and are relevant to recent experiments on Bose-Einstein condensates freely expanding past obstacles.

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Two-dimensional periodic waves in supersonic flow of a Bose–Einstein condensate

Cited by 17 publications

References 16 publications

Oblique Spatial Dispersive Shock Waves in Nonlinear Schrödinger Flows

Oblique Spatial Dispersive Shock Waves in Nonlinear Schrödinger Flows

Dispersive shock waves and modulation theory

Two-dimensional supersonic nonlinear Schrödinger flow past an extended obstacle

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