A. We prove the existence of highest, cusped, periodic travelling-wave solutions with exact and optimal α-Hölder continuity in a family of negative-order fractional Korteweg-de Vries equations of the formfor every α ∈ (0, 1) with homogeneous Fourier multiplier |D| −α . We tackle nonlinearities n(u) of the type |u| p or u|u| p−1 for all real p > 1, and show that when n is odd, the waves also feature antisymmetry and thus contain inverted cusps. The analysis for nonsmooth n is applicable to other negative-order nonlocal equations. Both the construction of highest antisymmetric waves and the regularisation of nonsmooth terms to an analytic bifurcation setting, we believe are new in this context.
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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