fact for integrable completely monotone functions). A closed formula for the periodised kernel is given in Corollary \[cor:periodic formula\].

In Section \[sec:nodal\] we prove some general lemmas about solutions of , whereof the most important to us is Theorem \[thm:nodal\], which establishes the nodal properties of solutions along the main bifurcation branch to be constructed. As it turns out, satisfies a maximum principle (touching lemma), making it resemblant of an elliptic equation. The nodal properties are essential in avoiding the closed-loop alternative in the global bifurcation analysis, but they also give information about the waves in their own right.

Section \[sec:singularity\] is the main part of the paper, in the sense that it contains the a priori analysis the highest wave. It is also the most technical, making use of both Besov spaces and, mostly, of integral estimates adapted for the assumed optimal regularity of the wave. Since $K(x) \sim |x|^{-1/2}$ and we expect $\frac{\mu}{2} - \varphi \sim |x|^{1/2}$, both relations for small values of $x$, one difficulty is that the integral $\int K^\prime(y) (\frac{\mu}{2} - \varphi(y))\,{{\mathrm d}}y$ diverges exactly at the expected regularity; another is that the point where $\varphi = \frac{\mu}{2}$ must be treated separately from other points. 