Orbital Bifurcations and Shoaling of Cnoidal Waves

We study the parameter space of cnoidal waves -- the periodic solitons of the Korteweg-de Vries equation -- from the perspective of Virasoro coadjoint orbits. The monodromy method familiar from inverse scattering implies that many, but not all, of these solitons are conformally equivalent to uniform field configurations (constant coadjoint vectors). The profiles that have no uniform representative lie in Lam\'e band gaps and are separated from the others by bifurcation lines along which the corresponding orbits change from elliptic to hyperbolic. We show that such bifurcations can be produced by shoaling: wave profiles become non-uniformizable once their pointedness parameter crosses a certain critical value (which we compute). As a by-product, we also derive asymptotic relations between the pointedness and velocity of cnoidal waves along orbital level curves.