Periodic H\"older waves in a class of negative-order dispersive equations

We prove the existence of highest, cusped, periodic travelling-wave solutions with exact and optimal $\alpha$-H\"older continuity in a class of fractional negative-order dispersive equations of the form \begin{equation*} u_t + (| \mathrm{D} |^{- \alpha} u + n(u) )_x = 0 \end{equation*} for every $\alpha \in (0, 1)$ with homogeneous Fourier multiplier $| \mathrm{D} |^{ - \alpha}$. 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. Tools involve detailed pointwise estimates in tandem with analytic global bifurcation, where we resolve the issue with nonsmooth $n$ by means of regularisation. We believe that both the construction of highest antisymmetric waves and the regularisation of nonsmooth terms to an analytic bifurcation setting are new in this context, with direct applicability also to generalised versions of the Whitham, the Burgers--Poisson, the Burgers--Hilbert, the Degasperis--Procesi, the reduced Ostrovsky, and the bidirectional Whitham equations.