# Orbital Bifurcations and Shoaling of Cnoidal Waves

## TL;DR

This paper explores the bifurcations and shoaling effects in cnoidal waves, linking wave profile transformations to conformal equivalence and orbital dynamics within the Korteweg-de Vries framework.

## Contribution

It introduces a novel analysis of cnoidal wave bifurcations using Virasoro coadjoint orbits and computes the critical shoaling parameter for wave profile changes.

## Key findings

- Bifurcation lines separate elliptic and hyperbolic orbits.
- Shoaling induces non-uniformizable wave profiles.
- Derived asymptotic relations between pointedness and wave velocity.

## Abstract

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.