Determination of boundaries of unsteady oscillatory zone in asymptotic solutions of non-integrable dispersive wave equations

El, Gennady; Khodorovskii, V.V.; Tyurina, A. V.

doi:10.1016/j.physleta.2003.09.060

Cited by 19 publications

(70 citation statements)

References 12 publications

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“…Remarkably, these limiting integrals can be expressed in terms of the linear dispersion relation of the original system, and one of its "dispersionless" nonlinear characteristic velocities. The corresponding analysis has been recently described in [43], [44],…”

Section: Unsteady Undular Bore Transition a General Constructionmentioning

confidence: 99%

“…Then the speeds of the undular bore edges s ± can be determined by the general expressions derived in [43], [44], [45]. Below we briefly outline how this method applies to the SG system (1).…”

Section: Undular Bore Location and Lead Solitary Wave Amplitudementioning

confidence: 99%

“…(17), (18) in the limit k → 0. This difficulty is bypassed in the papers of [43], [44], [45] by the introduction of the conjugate wavenumber…”

Section: Undular Bore Location and Lead Solitary Wave Amplitudementioning

confidence: 99%

“…However, unlike the situation for the modulation equations associated with completely integrable equations, the modulation equations associated with non-integrable wave equations do not possess the Riemann invariant structure. This feature is a serious obstacle to their investigation.However, recently an analytic approach has been developed in this case [43], [44], [45], which allows for the determination of a set of "transition conditions" across an unsteady undular bore, and which does not require the presence of the Riemann invariant structure for the Whitham modulation equations. This approach takes advantage of some specific properties of the Whitham modulation equations connected with their origin as certain 6 averages of exact conservation laws, and so allows for exact reductions in the zero-amplitude and zero-wavenumber limits.…”

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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: Unsteady Undular Bore Transition a General Constructionmentioning

confidence: 99%

Section: Undular Bore Location and Lead Solitary Wave Amplitudementioning

confidence: 99%

“…(17), (18) in the limit k → 0. This difficulty is bypassed in the papers of [43], [44], [45] by the introduction of the conjugate wavenumber…”

Section: Undular Bore Location and Lead Solitary Wave Amplitudementioning

confidence: 99%

mentioning

confidence: 99%

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Unsteady undular bores in fully nonlinear shallow-water theory

2006

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“…When the governing nonlinear wave equation does not have an inverse scattering solution it is usually impossible to set the modulation equations in Riemann invariant form, so that the undular bore solution can be determined. In this case, a general method has been developed which can determine the solitary waves at the leading edge and the linear waves at the trailing edge of the bore [19,20,21]. However, this method relies on the nonlinear wave equation being hyperbolic outside of the bore region.…”

Section: Introductionmentioning

confidence: 99%

Approximate Techniques for Dispersive Shock Waves in Nonlinear Media

Marchant

Smyth

2012

J. Nonlinear Optic. Phys. Mat.

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Many optical and other nonlinear media are governed by dispersive, or diffractive, wave equations, for which initial jump discontinuities are resolved into a dispersive shock wave. The dispersive shock wave smooths the initial discontinuity and is a modulated wavetrain consisting of solitary waves at its leading edge and linear waves at its trailing edge. For integrable equations the dispersive shock wave solution can be found using Whitham modulation theory. For nonlinear wave equations which are hyperbolic outside the dispersive shock region, the amplitudes of the solitary waves at the leading edge and the linear waves at the trailing edge of the dispersive shock can be determined. In this paper an approximate method is presented for calculating the amplitude of the lead solitary waves of a dispersive shock for general nonlinear wave equations, even if these equations are not hyperbolic in the dispersionless limit. The approximate method is validated using known dispersive shock solutions and then applied to calculate approximate dispersive shock solutions for equations governing nonlinear optical media, such as nematic liquid crystals, thermal glasses and colloids. These approximate solutions are compared with numerical results and excellent comparisons are obtained. Abstract Many optical and other nonlinear media are governed by dispersive, or diffractive, wave equations, for which initial jump discontinuities are resolved into a dispersive shock wave. The dispersive shock wave smooths the initial discontinuity and is a modulated wavetrain consisting of solitary waves at its leading edge and linear waves at its trailing edge. For integrable equations the dispersive shock wave solution can be found using Whitham modulation theory. For nonlinear wave equations which are hyperbolic outside the dispersive shock region, the amplitudes of the solitary waves at the leading edge and the linear waves at the trailing edge of the dispersive shock can be determined. In this paper an approximate method is presented for calculating the amplitude of the lead solitary waves of a dispersive shock for general nonlinear wave equations, even if these equations are not hyperbolic in the dispersionless limit. The approximate method is validated using known dispersive shock solutions and then applied to calculate approximate dispersive shock solutions for equations governing nonlinear optical media, such as nematic liquid crystals, thermal glasses and colloids. These approximate solutions are compared with numerical results and excellent comparisons are obtained.

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