Camassa-Holm Equations with an internal symmetry

In the first part of my talk, I will revisit my work on the scalar Camassa–Holm equation, which will set the stage for the second part. There, I will outline a construction of spinor analogs of the Camassa–Holm equation. In essence, each orthogonal group gives rise to a Camassa–Holm–type equation with intricate internal dynamics. I will motivate this generalization using spectral deformations of the Euler–Bernoulli beam problem, which corresponds to the Clifford algebra on two generators with Minkowski signature. The dynamics of solutions of this Clifford extension are far more intricate than in the scalar case, a contrast I will illustrate with concrete examples. The talk is based on recent joint work with R. Beals and ongoing research with A. Hone and V. Novikov.