Decay of an initial discontinuity in the defocusing NLS hydrodynamics

El, Gennady; Geogjaev, V. V.; Gurevich, A.; Крылов, А. Л.

doi:10.1016/0167-2789(95)00147-v

Cited by 124 publications

(271 citation statements)

References 8 publications

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“…We consider the zero-amplitude reduction of the full modulation system with the built-in simplewave relationū = 2( √η −1), which is an exact integral of the ideal shallow-water equations (21) and which is consistent with the relationship (37) between u and η at the trailing edge of the undular bore,…”

Section: Formation Of a Finite-amplitude Rear Wave Frontmentioning

confidence: 99%

“…This approach to the modulation dynamics was first proposed by Gurevich and Pitaevskii [19] for the KdV equation, and has proved to be very effective for description of weakly nonlinear unsteady undular bores (see for instance [17], [18], and [3]). The Gurevich-Pitaevskii formulation has been extended to other integrable models such as the defocusing nonlinear Schrödinger equation [20], [21], [22] , the BenjaminOno equation [23] and the Kaup-Boussinesq system [24], [53]. The original GurevichPitaevskii formulation makes essential use of the availability of the Riemann invariants for the modulation system.…”

Section: Unsteady Undular Bore Transition a General Constructionmentioning

confidence: 99%

“…This remarkable fact was first established by Whitham [7] for the third-order KdV modulation system, and then generalised to many other completely integrable systems (see [50] for a simple method of obtaining the Whitham system in Riemann form for a large class of integrable dispersive equations belonging to Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy). The presence of the Riemann invariants dramatically simplifies further analysis of the modulation equations, and makes readily available many important particular solutions (see [21], [22], [24] for instance). However, for non-integrable equations such a structure is typically not available, which makes the corresponding analysis of the modulation system far more complicated.…”

Section: Whitham Modulation Equationsmentioning

confidence: 99%

“…Significantly in this integrable case, the modulation equations can be obtained in the Riemann invariant form. Later, this Gurevich-Pitaevskii theory has been generalised to other integrable equations such as the defocusing nonlinear Schrödinger equation [20], [21], [22], the Benjamin-Ono equation [23] and the Kaup-Boussinesq system [24]. All these cited works make essential use of the Riemann form of the modulation system, a feature associated with the complete integrability of the original equation.…”

Section: Introductionmentioning

confidence: 99%

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Unsteady undular bores in fully nonlinear shallow-water theory

2006

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We consider unsteady undular bores for a pair of coupled equations of Boussinesqtype which contain the familiar fully nonlinear dissipationless shallow-water dynamics and the leading-order fully nonlinear dispersive terms. This system contains one horizontal space dimension and time and can be systematically derived from the full Euler equations for irrotational flows with a free surface using a standard long-wave asymptotic expansion. In this context the system was first derived by Su and Gardner. It coincides with the one-dimensional flat-bottom reduction of the Green-Naghdi 1 system and, additionally, has recently found a number of fluid dynamics applications other than the present context of shallow-water gravity waves. We then use the Whitham modulation theory for a one-phase periodic travelling wave to obtain an asymptotic analytical description of an undular bore in the Su-Gardner system for a full range of "depth" ratios across the bore. The positions of the leading and trailing edges of the undular bore and the amplitude of the leading solitary wave of the bore are found as functions of this "depth ratio". The formation of a partial undular bore with a rapidly-varying finite-amplitude trailing wave front is predicted for "depth ratios" across the bore exceeding 1.43. The analytical results from the modulation theory are shown to be in excellent agreement with full numerical solutions for the development of an undular bore in the Su-Gardner system. I INTRODUCTIONIn shallow water, the transition between two different basic states, each characterized by a constant depth and horizontal velocity, is usually referred to as a bore. For sufficiently large transitions, the front of the bore is often turbulent, but as noted in the classical work of Benjamin and Lighthill [1], transitions of moderate amplitude are accompanied by wave trains without any wave breaking, and are hence called undular bores. Well-known examples are the bores on the River Severn in England and the River Dordogne in France.Undular bores also arise in other fluid flow contexts; for instance they can occur as internal undular bores in the density-stratified waters of the coastal ocean (see, for instance, [2], [3]), and as striking wave-forms with associated cloud formation in the atmospheric boundary layer (see, for instance, [4], [5]). They can also arise in many other physical contexts, and in plasma physics for instance, are usually called collisionless shocks.The classical theory of shallow-water undular bores was initiated by Benjamin and Lighthill in [1]. It is based on the analysis of stationary solutions of the Kortewegde Vries (KdV) equation modified by a small viscous term [6]. Subsequent approaches to the same problem have been usually based on the Whitham modulation theory (see 2 [7], [8]), appropriately modified by dissipation; this allows one to study analytically the development of an undular bore to a steady state (see [9], [10], [11]). Most recently, this approach was used in [12] to study the development of an u...

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Section: Formation Of a Finite-amplitude Rear Wave Frontmentioning

confidence: 99%

Section: Unsteady Undular Bore Transition a General Constructionmentioning

confidence: 99%

Section: Whitham Modulation Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Unsteady undular bores in fully nonlinear shallow-water theory

2006

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“…Classification of possible wave structures in the NLS equation theory was given in Refs. [7,8], and it provides the theoretical basis for calculation of characteristic parameters of such experiments as that of Ref. [6].…”

Section: Introductionmentioning

confidence: 99%

Riemann problem for the photon fluid: Self-steepening effects

Ivanov

Камчатнов

2017

Phys. Rev. A

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We consider the Riemann problem of evolution of initial discontinuities for the photon fluid propagating in a normal dispersion fiber with account of self-steepening effects. The dynamics of light field is described by the nonlinear Schrödinger (NLS) equation with self-steepening term appearing due to retardation of the fiber material response to variations of the electromagnetic signal. It is shown that evolution dynamics in this case is much richer than that for the NLS equation. Complete classification of possible wave structures is given for all possible jump conditions at the discontinuity.

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Dispersive Shock Waves in Reorientational and Other Optical Media

Marchant

2012

Nematicons

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Decay of an initial discontinuity in the defocusing NLS hydrodynamics

Cited by 124 publications

References 8 publications

Unsteady undular bores in fully nonlinear shallow-water theory

Unsteady undular bores in fully nonlinear shallow-water theory

Riemann problem for the photon fluid: Self-steepening effects

Dispersive Shock Waves in Reorientational and Other Optical Media

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