Periodic Hölder waves in a class of negative-order dispersive equations

Abstract

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.

Highest Cusped Waves for the Burgers-Hilbert equation

“…Hildrum and Xue prove a similar result for another class of equations, including the (homogeneous) fractional KdV equations for −1 < α < 0 [24]. The results in [18,7,48,24] are all based on global bifurcation arguments, bifurcating from the constant solution and proving that the branch must end in a highest wave which is not smooth at its crest. The proof of the convexity of the highest cusped wave for the Whitham equation in [20] uses a completely different approach.…”

“…In this section we reduce the problem of proving Theorem 1.1 to proving the existence of a fixed point for a certain operator. We start with the following lemma which motivates the notion of highest wave (see also [24,Theorem 3.4]) Lemma 2.1. Let ϕ ∈ C 1 be a nonconstant, even solution of (3) which is nondecreasing on (−π, 0), then ϕ > 0 and ϕ < c on (−π, 0).…”

“…. The bulk of the work is for the second interval, it needs to be split into more than 2 24 subintervals and the final enclosure for the maximum is [0.0003448 ± 4.98 We are able to compute Taylor expansions of F (x) in both the asymptotic and non-asymptotic case as long as the subinterval doesn't contain zero, this allows us to use the better version of the algorithm for enclosing the maximum and fall back to the naive version for the subintervals containing zero. We use a Taylor expansion of degree 4 in all cases.…”

“…Which for s − 1 > 0 and x ∈ (0, π) is positive, C s (x) is hence decreasing. For s < 1 we use equation (24) together with [15, Eq. 25.11.25] to get…”

“…In all the above cases we could reduce the problem of enclosing the Clausen functions on x to evaluating them on, a subset of, x = 0, x = x (or −x if x < 0), x = x, x = x 0 (the midpoint) and x = π. In all cases except x = 0 we have 0 < x < 2π so we can use (23) and (24).…”