is then straightforward to find a subsequence of waves converging to a solution with $\max \varphi = \mu/2$, and we use elliptic properties of the equation to rule out the possibility of the wave speed $\mu$ vanishing in the limit.

![The highest wave found in Theorem \[thm:main\]. This travelling-wave solution of is obtained as a limit along the main global bifurcation curve established in Theorem \[thm:global\_whit\]. By construction, the solution is $P$-periodic, even, and strictly increasing on the interval $(-P/2,0)$, satisfying $\varphi(0) = \frac{\mu}{2}$. As proved in Theorem \[thm:regularity II\], it is furthermore smooth away from any crest, and obtains its optimal Hölder regularity $C^{1/2}({\mathbb R})$ exactly at the crest. []{data-label="fig:highest"}](highest.pdf){width="0.5\linewidth"}

For an in-depth analysis of the resulting limiting wave a detailed study of the integral equation appears to be unavoidable. Functional-analytic arguments provides one with $C^\alpha$-regularity for any $\alpha < 1/2$, but not better. To improve our estimates we use several differing ways of expressing $\varphi(x+h) - \varphi(x-h)$, which makes it possible to move first- and second-order differences between $K$ and $\varphi$ in the integrals that appear. We first move second-order differences to $K$ and use the $C^\alpha$-regularity of $\varphi$, $\alpha < 1/2$, to get $C^{1/2}$-regularity exactly at the 