Short-wave dynamics in the Euler equations

Manna, M. A.; Neveu, A.

doi:10.1088/0266-5611/17/4/317

Cited by 9 publications

(20 citation statements)

References 33 publications

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“…Indeed, the linear dispersion relation (2) can be regarded as a Padé approximation of the exact expression (ω 0 − ku 0 ) 2 = k tanh kη 0 following from the Euler equations, whereas the KdV dispersion relation is a less accurate Taylor series approximation. Applications of modifications of system (1) to the description of the nonlinear dynamics of short waves have been considered in [40], [41]. …”

Section: Undular Bore Location and Lead Solitary Wave Amplitudementioning

confidence: 99%

“…With some modifications it has also been used for modelling short surface waves governed by the Euler equations [40] and for capillary-gravity waves [41]. The "two-layer" generalisation of the same onedimensional fully nonlinear shallow-water system for large amplitude internal wave propagation has been obtained in [37] using asymptotic expansions and recently re-derived in Thus, these equations (provided they are used in a consistent asymptotic approximation) represent an important mathematical model for understanding general properties of large-amplitude dispersive long waves.…”

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: Undular Bore Location and Lead Solitary Wave Amplitudementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Unsteady undular bores in fully nonlinear shallow-water theory

2006

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“…Therefore, it seems to be important to capture the physical phenomena described by these equations. In the context of surface gravity waves, a few works along this line are found, for example, in [12,18,28].. where v = du/dX and d is an integration constant. In the following analysis, we consider the case c < 0 for which the periodic wave solutions would exist if the constant d satisfies certain condition.…”

Section: Discussionmentioning

confidence: 99%

Parametric solutions of the generalized short pulse equations

Matsuno

2020

J. Phys. A: Math. Theor.

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We consider three novel PDEs associated with the integrable generalizations of the short pulse equation classified recently by Hone et al (2018 Lett. Math. Phys. 108 927-947). In particular, we obtain a variety of exact solutions by means of a direct method analogous to that used for solving the short pulse equation. The main results reported here are the parametric representations of the multisoliton solutions. These solutions include cusp solitons, unbounded solutions with finite slope and breathers, as well as traveling periodic waves with cusp singularity. In addition, the smooth periodic traveling wave solutions are provided by employing phase plane analysis. Several new features of solutions are exhibited. As for non-periodic solutions, smooth breather solutions are of particular interest from the perspective of applications to real physical phenomena. The cycloid reduced from the periodic traveling wave with cusps is also worth remarking in connection with Gerstner's trochoidal solution in deep gravity waves. A number of works are left for future study, some of which will be addressed in concluding remarks.

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“…Its Lax pair and complete integrability were shown in [47]. The change of variable and of function (61) and (62) leading from (56) to (63) were introduced in the context of a systematic study of short-wave dynamics in long-wave model equations in references [48,49,50,51].…”

Section: Reduction To the Bullough-dodd Equationmentioning

confidence: 99%

An integrable evolution equation for surface waves in deep water

Kraenkel¹,

Leblond²,

Manna³

2013

J. Phys. A: Math. Theor.

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In order to describe the dynamics of monochromatic surface waves in deep water, we derive a nonlinear and dispersive system of equations for the free surface elevation and the free surface velocity from the Euler equations in infinite depth. From it, and using a multiscale perturbative methods, an asymptotic model for small-aspectratio waves is derived. The model is shown to be completely integrable. The Lax pair, the first conserved quantities as well as the symmetries are exhibited. Theoretical and numerical studies reveal that it supports periodic progressive Stokes waves which peak and break in finite time. Comparison between the limiting wave solution of the asymptotic model and classical irrotational results is performed.

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Short-wave dynamics in the Euler equations

Cited by 9 publications

References 33 publications

Unsteady undular bores in fully nonlinear shallow-water theory

Unsteady undular bores in fully nonlinear shallow-water theory

Parametric solutions of the generalized short pulse equations

An integrable evolution equation for surface waves in deep water

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