Reflections from solitary waves in channels of decreasing depth

Knickerbocker, C. J.; Newell, Alan C.

doi:10.1017/s0022112085001112

Cited by 25 publications

(24 citation statements)

References 11 publications

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“…Note that, by virtue of the large-radius scaling, the geometric contribution in the cylindrical coordinates first appears in (12), leaving (11) as the classical KdV equation. The various components described by equations (11)- (14) are represented schematically in Fig(2); see the descriptions that follow, where we give a brief outline of what can be deduced at each order.…”

Section: Resultsmentioning

confidence: 99%

“…The propagation of plane solitary waves over variable depth is now well understood ( [3,14,10,11,7]). In this problem it is shown that, starting from an initial solitary-wave solution of the KdV equation ( [12]), the propagation over a region of varying depth introduces a shelf directly behind the primary wave, a left-going (reflected) 'shelf' and another right-going 'shelf' (re-reflection).…”

Section: Introductionmentioning

confidence: 99%

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Propagation of axi-symmetric nonlinear shallow water waves over slowly varying depth

Killen

Johnson

2001

Mathematics and Computers in Simulation

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A problem in nonlinear water-wave propagation on the surface of an inviscid, stationary fluid is presented.The primary surface wave, suitably initiated at some radius, is taken to be a slowly evolving nonlinear cylindrical wave (governed by an appropriate Korteweg-de Vries equation); the depth is assumed to be varying in a purely radial direction.We consider a sech 2 profile at an initial radius (which is, following our scalings, rather large), and we describe the evolution as it propagates radially outwards. This initial profile was chosen because its evolution over constant depth is understood both analytically and numerically, even though it is not an exact solitary-wave solution of the cylindrical KdV equation. The propagation process will introduce reflected and re-reflected components which will also be described. The precise nature of these reflections is fixed by the requirements of mass conservation.The asymptotic results presented describe the evolution of the primary wave, the development of an outward shelf and also an inward (reflected) shelf. These results make use of specific depth variations (which were chosen to simplify the solution of the relevant equations), and mirror those obtained for the problem of 1-D plane waves over variable depth, although the details here are more complex due to the axi-symmetry.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Propagation of axi-symmetric nonlinear shallow water waves over slowly varying depth

Killen

Johnson

2001

Mathematics and Computers in Simulation

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“…Then, returning to the original (dimensionless) variables and coordinates, x, y and t, one may express Eqs. (35) and (36) as follows:…”

Section: A Unperturbed Soliton Solutionsmentioning

confidence: 99%

Hydrodynamics and two-dimensional dark lump solitons for polariton superfluids

et al. 2018

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We study a two-dimensional incoherently pumped exciton-polariton condensate described by an open-dissipative Gross-Pitaevskii equation for the polariton dynamics coupled to a rate equation for the exciton density. Adopting a hydrodynamic approach, we use multiscale expansion methods to derive several models appearing in the context of shallow water waves with viscosity. In particular, we derive a Boussinesq/Benney-Luke-type equation and its far-field expansion in terms of Kadomtsev-Petviashvili-I (KP-I) equations for right- and left-going waves. From the KP-I model, we predict the existence of vorticity-free, weakly (algebraically) localized two-dimensional dark-lump solitons. We find that, in the presence of dissipation, dark lumps exhibit a lifetime three times larger than that of planar dark solitons. Direct numerical simulations show that dark lumps do exist, and their dissipative dynamics is well captured by our analytical approximation. It is also shown that lumplike and vortexlike structures can spontaneously be formed as a result of the transverse "snaking" instability of dark soliton stripes.

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“…The efforts have then been directed at discovering all the wave components (reflected waves, etc.) which contribute, for example, to the overall mass balance; see [36,30,31,23]. It is a fairly straightforward exercise to analyse the even more general problem of an ambient shear flow which varies as the water flows in a variable-depth environment; the modifications of the classical KdV equation that are evident in equations (4.4) and (4.7) are combined in this new problem; see [25].…”

Section: Extending the Kdv Modelmentioning

confidence: 99%

The Classical Problem of Water Waves: a Reservoir of Integrable and Nearly-Integrable Equations

Johnson¹

2003

JNMP

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In this contribution, we describe the simplest, classical problem in water waves, and use this as a vehicle to outline the techniques that we adopt to analyse this particular approach to the derivation of soliton-type equations. The surprise, perhaps, is that such an apparently transparent set of equations (the Euler equation for an incompressible fluid, the equation of mass conservation and the simplest bottom and surface conditions) contains so may different -and important -equations.We will briefly show how such equations are generated, by carefully describing the asymptotic procedures that we adopt. We will then present a number of examples, some of which will be the familiar integrable equations (Korteweg-de Vries (KdV), Nonlinear Schrödinger (NLS)) and also variants of these which are, in a sense, nearlyintegrable. These will include, for example, a variable coefficient KdV equation, a nearly concentric KdV equation and higher-order NLS equation. Some of these results have significant consequences for water-wave propagation; for example, the existence of a KdV equation for arbitrary velocity profiles below the surface helps to explain why solitary waves are observed in nature.We conclude with a discussion of some of the more recent, exciting work on the rôle of the Camassa-Holm (CH) equation in water waves. We will also outline how, for this type of problem, a number of interesting and relevant variants of the CH equation can arise.

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Reflections from solitary waves in channels of decreasing depth

Cited by 25 publications

References 11 publications

Propagation of axi-symmetric nonlinear shallow water waves over slowly varying depth

Propagation of axi-symmetric nonlinear shallow water waves over slowly varying depth

Hydrodynamics and two-dimensional dark lump solitons for polariton superfluids

The Classical Problem of Water Waves: a Reservoir of Integrable and Nearly-Integrable Equations

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