Convexity of Whitham's highest cusped wave

Enciso, Alberto; Gómez-Serrano, Javier; Vergara, Bruno

doi:10.48550/arxiv.1810.10935

Cited by 9 publications

(20 citation statements)

References 18 publications

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“…After this breakthrough work, there have been an increasing number of studies on nonlocal equations using their approach. One of the earliest papers in this direction is to justify the convexity of the wave profile between consecutive crests using a computer-assisted proof [14]. A similar result as in [12] but for solitary waves was also obtained by Truong, Wahlén, and Wheeler [20] using a center manifold theorem approach.…”

Section: Introductionmentioning

confidence: 75%

Waves of maximal height for a class of nonlocal equations with inhomogeneous symbols

2020

Preprint

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Section: Introductionmentioning

confidence: 75%

Waves of maximal height for a class of nonlocal equations with inhomogeneous symbols

2020

Preprint

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“…Even more, in [10] it was shown that there exist cusped, periodic traveling-wave solutions to the Whitham equation, and that they have exact 1/2-Hölder regularity at crests -corresponding to the order of the dispersion in the equation. It was conjectured that such solutions are convex between cusps, which was confirmed by a computer-assisted proof in [12].…”

Section: Introductionmentioning

confidence: 78%

Highest waves for fractional Korteweg--De Vries and Degasperis--Procesi equations

Ørke¹

2022

Preprint

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We study traveling waves for a class of fractional Korteweg-De Vries and fractional Degasperis-Procesi equations with a parametrized Fourier multiplier operator of order −s ∈ (−1, 0). For both equations there exist local analytic bifurcation branches emanating from a curve of constant solutions, consisting of smooth, even and periodic traveling waves. The local branches extend to global solution curves. In the limit we find a highest, cusped traveling-wave solution and prove its optimal s-Hölder regularity, attained in the cusp.

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“…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].…”

Section: Introductionmentioning

confidence: 89%

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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Convexity of Whitham's highest cusped wave

Cited by 9 publications

References 18 publications

Waves of maximal height for a class of nonlocal equations with inhomogeneous symbols

Waves of maximal height for a class of nonlocal equations with inhomogeneous symbols

Highest waves for fractional Korteweg--De Vries and Degasperis--Procesi equations

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

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