show, periodic waves with a sharp crest. This was not clear. In fact, the operator $L$ is not only weaker, but *much* weaker than that of both the KdV equation and almost any other shallow water wave equation, so that the existence of solitary waves was, until recently [@EGW11], an open problem. So was wave breaking (see just below), and the existence of a highest, cusped, travelling wave.

Singularities in solutions of appear in two forms: in the form of wave breaking when the spatial derivative of a bounded solution blows up in the evolution problem, and in the form of a sharp crests for a travelling wave. Although the idea behind wave breaking was introduced already by Seliger [@0159.28502], the full details for the Whitham equation were settled much later, with [@0802.35002], [@MR1668586] and [@Hur15]. We, however, shall be concerned with steady waves.

In steady variables $\varphi(\tilde x) = u(x-\mu t)$ the Whitham equation takes the form $$\label{eq:steadywhitham}
-\mu \varphi + L\varphi +\varphi^2 = 0,$$ where the equation has been integrated once, and the constant of integration set to zero. There is no loss of generality in doing so, since the Galilean change of variables $$\varphi \mapsto \varphi + 