linear dispersion and nonlinear effects, so that one would have smooth periodic and solitary waves, but also the features of wave breaking and surface singularities. To accomplish that he considered the symbol $$m(\xi) = \sqrt{{\textstyle \frac{\tanh{\xi}}{\xi} }},$$ arising as the full frequency dispersion for linear gravity water waves on finite depth, and its inverse Fourier transform, $$\label{eq:K}
K(x) = \frac{1}{2\pi}\int_{\mathbb R}m(\xi) \exp(ix\xi)\,{{\mathrm d}}\xi.$$ If one denotes by $L \colon f \mapsto K \ast f$ the action by convolution with the kernel $K$, the *Whitham equation* is the nonlinear, nonlocal evolution equation $$\label{eq:whitham}
u_t + (Lu  + u^2)_x = 0.$$ While many shallow water-wave equations can be written in this form, their symbols are generally leading order approximations of the exact linear dispersion $m(\xi)$, and therefore behaves radically different for large frequencies $\xi$; a typical example is the Korteweg–de Vries equation, whose symbol $1 - \frac{1}{6}\xi^2$ consists of the two first terms in the Maclaurin series for $m(\xi)$. The goal of introducing the operator $L$, on the other hand, was to weaken the dispersion so as to allow also for solutions with singularities.

As it turns out, Whitham was correct: the equation features solitary waves, wave breaking, and, as we will 