Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities

Maehlen, Ola I. H.

doi:10.3934/dcds.2020174

Cited by 7 publications

(17 citation statements)

References 19 publications

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“…The above assumptions to some extent describe the results in terms of the behaviour of the dispersive symbol at infinity in (A2) and the order of its local extremum at the origin in (A3). This is in line with an ongoing program to describe the properties of solutions to nonlocal dispersive equations in terms of a quantifiable properties of their nonlocalities and nonlinearities, see for example [6,7,18] for investigations based on the same idea.…”

Section: Introductionsupporting

confidence: 66%

Enhanced existence time of solutions to evolution equations of Whitham type

Ehrnström,

Wang

2020

Preprint

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We show that Whitham type equations ut +uux −Lux = 0, where L is a general Fourier multiplier operator of order α ∈ [−1, 1], α = 0, allow for small solutions to be extended beyond their expected existence time. The result is valid for a range of quadratic dispersive equations with inhomogeneous symbols in the dispersive range given by α, and should be extendable to other equations of the same relative dispersive strength.

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Section: Introductionsupporting

confidence: 66%

Enhanced existence time of solutions to evolution equations of Whitham type

Ehrnström,

Wang

2020

Preprint

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“…where q parametrises size of a solitary wave in some sense. Implementation of the Lions principle in the spirit of [47] to a minimizing sequence provides us with solitary waves. In addition we analyse the long wave asymptotic of the obtained solutions following closely arguments of [32].…”

Section: Solitary Wave Solutions Of a Whitham-boussinesq Systemmentioning

confidence: 99%

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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“…In order to show that the nonlocal interaction disappears as k → ∞, one can introduce certain commutators and prove that their operator norms vanish[26]. Based on uniform continuity of ξ → m(ξ)/〈ξ〉 s , which holds automatically in our case, this is applicable for a large class of symbols.…”

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confidence: 98%

Solitary waves in dispersive evolution equations of Whitham type with nonlinearities of mild regularity

Hildrum

2020

Nonlinearity

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A. We show existence of small solitary and periodic traveling-wave solutions in Sobolev spaces H s , s > 0, to a class of nonlinear, dispersive evolution equations of the formwhere the dispersion L is a negative-order Fourier multiplier whose symbol is of KdV type at low frequencies and has integrable Fourier inverse K and the nonlinearity n is inhomogeneous, locally Lipschitz and of superlinear growth at the origin. This generalises earlier work by Ehrnström, Groves & Wahlén on a class of equations which includes Whitham's model equation for surface gravity water waves featuring the exact linear dispersion relation. Tools involve constrained variational methods, Lions' concentration-compactness principle, a strong fractional chain rule for composition operators of low relative regularity,and a cut-off argument for n which enables us to go below the typical s > 1 2 regime. We also demonstrate that these solutions are either waves of elevation or waves of depression when K is nonnegative, and provide a nonexistence result when n is too strong. Key words and phrases: solitary waves; Whitham-type equations; nonlinear dispersive equations. Mathematics Subject Classification (2010): 35A01; 35A15; 35Q35; 76B03; 76B15; 76B25.after integrating (1). 1 30 arXiv:1903.03354v2 [math.AP] 6 Jan 2020 2/30 S W Whitham KdV ξ = 1 − 1 6 ξ 2 KdV symbol +O ξ 4 and fig. 1, it is intuitively reasonable that Whitham's model should both perform better and on a wider range of wave numbers than the KdV equation.Unfortunately, the nonlocal, singular nature of L-due to m(ξ) 〈ξ〉 − 1 2 being inhomogeneous and decaying very slowly at infinity-seems to have prevented people from rigorously studying the Whitham equation until recently. Significant breakthrough in the last decade, however, has put the original Whitham equation, and also other full-dispersion models, in the spotlight, beginning with the existence of periodic traveling waves by Ehrnström and Kalisch [9] in 2009 and solitary-wave solutions by Ehrnström, Groves and Wahlén [8] in 2012; see also [30]. Research has furthermore confirmed Whitham's conjectures for qualitative wave breaking (bounded wave profile with unbounded slope) in finite time [16] and the existence of highest, cusp-like solutions [10, 12]-now known to also have a convex profile between the stagnation points [13].Additional analytical and numerical results for the Whitham equation include modulational instability of periodic waves [17,29], local well-posedness in Sobolev spaces H s , s > 3 2 , for both solitary and periodic initial data [7,11,19], non-uniform continuity of the data-to-solution map [1], symmetry and decay of traveling waves [3], analysis of modeling properties, dynamics and identification of scaling regimes [19], and wave-channel experiments and other numerical studies [2,5,18,32].In total, these investigations have demonstrated the potential usefulness of full-dispersion versions of traditional shallow-water models.1.2 Assumptions and main results. In this paper we contribute to the longstanding mathema...

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Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities

Cited by 7 publications

References 19 publications

Enhanced existence time of solutions to evolution equations of Whitham type

Enhanced existence time of solutions to evolution equations of Whitham type

The Whitham equation for hydroelastic waves

Solitary waves in dispersive evolution equations of Whitham type with nonlinearities of mild regularity

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