a generic nonlinearity and such weak dispersion, see [@LannesSaut13] and [@MR3188389]. As described above, waves with sufficiently large inclination will eventually break, and numerical data indicates that the form of breaking waves mimics that of the highest wave constructed in this paper [@MR3317254]. The results presented in this paper are in turn based on [@EK08] and [@EK11], in which global branches of periodic, but smooth, periodic solutions were analytically constructed and numerically investigated.

The outline of our investigation is as follows. In Section \[sec:K\] we inspect the integral kernel $K$ corresponding to the symbol $m(\xi)$. Although some of our results are valid for general completely monotone functions, the results with most consequence for our further investigation are Propositions \[prop:g is Stieltjes\] and \[prop:Whitham formula\], where we prove that $K$ is completely monotone—meaning that all its odd derivatives are negative on a half-line, and contrariwise for the even derivatives—and give a closed formula for it. In Section \[sec:Kp\] we continue the study of the integral operator $L$, now for the periodised integral kernel $K_P$, and give some useful properties of $L$ in general. Interestingly, $K_P$ is completely monotone as well, on a half-period (this is not a coincidence, but a general 