Existence of a highest wave in a fully dispersive two-way shallow water model

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

“…The highest, periodic traveling-wave solution for the Whitham equation is exactly C 1 2 -Hölder continuous at its crests; thus exhibiting exactly half the regularity of the highest wave for the Euler equations. In a subsequent paper, Ehrnström, Johnson, and Claasen [10] studied the existence and regularity of a highest wave for the bidirectional Whitham equation incorporating the full Euler dispersion relation leading to a nonlocal equation with cubic nonlinearity and a Fourier multiplier with symbol mpkq " tanhpkq k . The question addressed in [10] is whether this equation gives rise to a highest, periodic, traveling wave, which is peaked (that is, whether it has a corner at each crest), such as the corresponding solution to the Euler equations?…”

“…In a subsequent paper, Ehrnström, Johnson, and Claasen [10] studied the existence and regularity of a highest wave for the bidirectional Whitham equation incorporating the full Euler dispersion relation leading to a nonlocal equation with cubic nonlinearity and a Fourier multiplier with symbol mpkq " tanhpkq k . The question addressed in [10] is whether this equation gives rise to a highest, periodic, traveling wave, which is peaked (that is, whether it has a corner at each crest), such as the corresponding solution to the Euler equations? Overcoming the additional challenge of the cubic nonlinearity, the authors in [10] follow a similar approach as implemented for the Whitham equation in [12] and prove that the highest wave has a singularity at its crest of the form |x logp|x|q|; thereby still being a cusped wave.…”

“…The question addressed in [10] is whether this equation gives rise to a highest, periodic, traveling wave, which is peaked (that is, whether it has a corner at each crest), such as the corresponding solution to the Euler equations? Overcoming the additional challenge of the cubic nonlinearity, the authors in [10] follow a similar approach as implemented for the Whitham equation in [12] and prove that the highest wave has a singularity at its crest of the form |x logp|x|q|; thereby still being a cusped wave. Concerning a different model equation arising in the context of shallow-water equations, Arnesen [3] investigated the existence and regularity of a highest, periodic, traveling-wave solution for the Degasperis-Procesi equation.…”

“…The advantage of this nonlocal approach relies not only in the fact that it can be applied to various equations of local and nonlocal type, but in particular, that it is suitable to study entire families of equations simultaneously; thereby providing an insight into the interplay between a certain nonlinearity and varying order of linearity. The main novelty in our work relies upon implementing the approach used in [12,10,3] for equations exhibiting homogeneous symbols. For a homogeneous symbol, the associated convolution kernel can not be identified with a positive, decaying function on the real line.…”

Convexity of Whitham's highest cusped wave

“…In this paper we will be concerned with the latter, whose existence was conjectured some forty years ago by Whitham [23] and established by Ehrnström and Wahlén [13] just recently. Interestingly, if one replaces the Whitham equation by a related fully dispersive model that contains both branches of the full Euler dispersion relation instead of just one, non-smooth traveling waves have been found too [10], but the solutions are in C α for all α < 1.…”