Solitary wave solutions to a class of Whitham–Boussinesq systems

Nilsson, Dag; Wang, Yuexun

doi:10.1007/s00033-019-1116-0

Cited by 21 publications

(15 citation statements)

References 29 publications

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“…In addition we analyse the long wave asymptotic of the obtained solutions following closely arguments of [32]. Thus we complement other results on solitary wave existence for the fully dispersive bidirectional models [32,50].…”

Section: Solitary Wave Solutions Of a Whitham-boussinesq Systemmentioning

confidence: 61%

The Whitham equation for hydroelastic waves

Dinvay

Kalisch

Moldabayev³

et al. 2019

Applied Ocean Research

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The PhD project concerns the surface water wave theory. Liquid is presented as a three or two dimensional layer bounded from below by a rigid horizontal bottom. Above it can have either a free surface or an elastic layer such as an ice cover, for example. The fluid flow is considered to be inviscid, incompressible and irrotational. The flow is described by the Euler equations with some appropriate boundary conditions. Following Lagrange we assume that at the bottom liquid has zero vertical velocity component and at the free surface there is a constant atmospheric pressure. The first condition is very natural meaning that water cannot penetrate through the bottom and is always attached to it making no cavities. The second condition is based on the fact that the air fluctuations are negligible and the pressure is defined by the weight of the atmosphere. Note that it also assumed that the pressure is changing continuously in the air-water media. The latter can be modified in different ways. For example, assuming a strong capillarity effect one can assume that the liquid pressure at the top is proportional to the surface curvature. However, it turns out that this effect is more important in water tanks than in the ocean. Another more relevant modification for the real world is the modelling of ice cover. In such situation the surface tension is caused by deformation forces in the ice. The latter is assumed to be elastic. Solving the Euler equations with the mentioned boundary conditions provides with the complete description of the flow dynamics. However, in many situations it is more important to know only the surface time evolution, whereas solving the full problem may be too demanding. Different nonlinear dispersive wave equations, such as the Korteweg-de Vries equation, allow to approximate the free surface dynamics without providing with the complete description of the fluid motion below the surface. There are several ways of testing the validity of these models. The two most important of them are tests on well-posedness and travelling wave existence. A relevant model should at least reflect adequately these two properties. In this work we are mainly concerned with the fully dispersive bi-directional extensions of the mentioned KdV equation. Still being simple toy models they are believed to be better approximations to the full Euler equations. Moreover, they allow two directional water motion for the two dimensional liquid layer, whereas the scalar KdV equation describes the waves traveling in one direction. A recent interest in such models was caused by discovery of certain phenomena in the Whitham equation that is a direct fully dispersive one-directional extension of the KdV equation. Among them are solitary waves, the existence of a wave of greatest height predicted by Stokes, the existence of shocks and modulational instability of steady periodic waves. A natural step forward is to try to find an adequate two-directional extension of the scalar Whitham equation displaying the same phenomena or some other w...

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Section: Solitary Wave Solutions Of a Whitham-boussinesq Systemmentioning

confidence: 61%

The Whitham equation for hydroelastic waves

Dinvay

Kalisch

Moldabayev³

et al. 2019

Applied Ocean Research

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“…The bidirectional Whitham equation (1.1) is mathematically interesting because of its weak dispersion, and contains a logarithmically cusped wave of greatest height [9] and solitary waves [18]. Experiments and numerical results indicate surprisingly good modelling properties for this model, as well as for several other 'Whitham-like' equations and systems, see [4,5,19].…”

Section: Introduction and Main Resultsmentioning

confidence: 91%

A note on well-posedness of bidirectional Whitham equation

Pei

Wang

2019

Applied Mathematics Letters

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“…with m(ξ ) = tanh(ξ )/ξ , which has taken a vast of attractions (we refer to [7] for a fairly complete list of references). There have been several investigations on the system (1.1) with m(ξ ) = tanh(ξ )/ξ [which corresponds to a = −1 in (1.3)]: local well-posedness [13,18], a logarithmically cusped wave of greatest height [6], existence of solitary wave solutions [17], and numerical results [3,4,21]. We should point out that the assumption in [13,18] on the initial surface elevation η 0 ≥ C > 0 is nonphysical, which only yields the well-posedness in homogeneous Sobolev spaces, however the Cauchy problem of (1.1) by this choice of symbol is probably ill-posed for negative initial surface elevation (see a heuristic argument in [13]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Well-Posedness to the Cauchy Problem of a Fully Dispersive Boussinesq System

Wang

2020

J Dyn Diff Equat

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This work concerns the local well-posedness to the Cauchy problem of a fully dispersive Boussinesq system which models fully dispersive water waves in two and three spatial dimensions. Our purpose is to understand the modified energy approach (Kalisch and Pilod in Proc Am Math Soc 147:2545-2559, 2019) in a different point view by utilizing the symmetrization of hyperbolic systems which produces an equivalent modified energy. Keywords Well-posedness • Dispersive Boussinesq system • Symmetrization Mathematics Subject Classification 76B15 • 76B03 • 35S30 in which t ∈ R, x = (x 1 , x 2) ∈ R 2 and the unknowns η ∈ R, v = (v 1 , v 2) ∈ R 2. The operator K is a Fourier multiplier with symbol m ∈ S a ∞ (R d), d = 1 or 2 for some a ∈ R\{0}, Y.W. acknowledges the support by Grants Nos. 231668 and 250070 from the Research Council of Norway.

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Solitary wave solutions to a class of Whitham–Boussinesq systems

Cited by 21 publications

References 29 publications

The Whitham equation for hydroelastic waves

The Whitham equation for hydroelastic waves

A note on well-posedness of bidirectional Whitham equation

Well-Posedness to the Cauchy Problem of a Fully Dispersive Boussinesq System

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