more so as the equation easily eludes any first attempts at obtaining such a cusped, highest, wave.

Even though the kernel $K$ in is real, even, and smooth for all $x \neq 0$ with derivates of rapid decay, as made precise in Proposition \[prop:decay\], Proposition \[prop:kernel decomposition\] and Corollary \[cor:improved asymptotics\], it is also singular at the origin, causing nontrivial problems when one wants to analyse it (to come to any conclusion, Whitham in his further analysis replaced the exact kernel $K$ with a continuous ‘approximation’ of it, something that however changes the weak dispersion of the equation , and interchanges it with the Burgers–Poisson equation that does not feature cusps). We approach $K$ by investigating the signs of its derivatives, taking a route via complex analysis and the theory of completely monotone functions. As it turns out, the kernel $K$ can be understood via both the theory of Stieltjes functions and the theory of positive definite functions, depending on whether one considers the Laplace or the Fourier transform, respectively. As a by-product of our study we obtain a closed formula for the kernel $K$ in physical space, as well as for its periodisation. It is worth noting that $K$ 