Henry P. McKean Jr. and Integrable Systems

Abstract

Henry McKean’s first contribution to integrable systems appeared in 1975 [15]. Over a period of more than 30 years since, in some 50 papers, he has explored integrable systems from uniquely original points of view. His selecta could have included the pioneering work, with Airault and Moser, on the time-dynamics of poles of meromorphic solutions of KdV [2]; or the series on invariant measures for wave equations, integrable or otherwise; or one of the seminal papers with Trubowitz [12, 13] that created a theory of infinite-genus hyperelliptic curves (= Riemann surfaces) and infinite-dimensionalJacobian varieties, which they applied to KdV under periodic boundary conditions, or subsequent extensions of that theory motivated by the desire to understand all the iconic integrable partial differential equation on the circle.

Appendix: Hamiltonian Mechanics

The prototypical Hamiltonian system in Euclidean space has the form

$$\displaystyle{ \dot{\mathbf{x}} = P\,\mathrm{grad}\,H(\mathbf{x}),\ \ \mathbf{x} \in \mathbb{R}^{N}, }$$

(A-1)

where P is a constant, skew-symmetric linear operator called Poisson operator and H is a given function called the Hamiltonian of (A-1). For simplicity, we assume P to be invertible, which forces N to be even, say N = 2n.

The vector field defined by the right side of (A-1) is written $\mathbb{X}_{H}$. The commutator of Hamiltonian vector fields is again Hamiltonian, $[\mathbb{X}_{H}, \mathbb{X}_{G}] = \mathbb{X}_{F}$, where F, called Poisson bracket of G and H, is given by the dot product

$$\displaystyle{F =\{ G,H\}:=\mathrm{ grad}\,G \cdot P\,\mathrm{grad}\,H.}$$

When {G, H} = 0, one says that G, H are in involution or Poisson commute. Equivalently, $[\mathbb{X}_{G}, \mathbb{X}_{H}] = 0$, which also implies that G is constant along integral curves of $\mathbb{X}_{H}$, and vice versa.

Suppose C ₁, …, C _n are (independent) functions in involution; there can be no more because their gradients span an isotropic subspace of P. The commuting vector fields $\mathbb{X}_{C_{j}}$ leave the n-dimensional level manifolds²⁵ $\mathbb{J}(\mathbf{c}):= \cap _{j}\{C_{j} = c_{j}\}$ invariant. If these are compact they must be tori. After a choice of origin, the flows of $\mathbb{X}_{C_{j}}$ define a transformation group (t ₁, …, t _n ) ↦ x(t ₁, …, t _n ) on each torus.

On an open set filled by n-dimensional tori $\mathbb{J}(\mathbf{c})$ one can introduce special coordinates, the action-angle coordinates, also called action-angle variables, $Z_{j},\theta _{j}$. The actions Z _j are functions of the c _k and label the tori, and the angles go around the cycles of the torus. These coordinates are canonical, meaning that all Poisson brackets vanish except for $\{Z_{j},\theta _{1}\} = 1$.

If a function H depends only on the actions Z _j , the Hamiltonian system takes the canonical form

$$\displaystyle{ \dot{\theta }_{j} = \frac{\partial H} {\partial Z_{j}},\ \dot{Z}_{j} = -\frac{\partial H} {\partial \theta _{j}} = 0, }$$

(A-2)

and since $\dot{\theta }_{j}$ is a constant, call it ω _j , the solution is

$$\displaystyle{Z_{j}(t) \equiv Z_{j}(0),\ \theta _{j}(t) =\omega _{j} \cdot t +\theta _{j}(0).}$$

Thus, in action-angle coordinates, the integral curves of the vector field $\mathbb{X}_{\tilde{H}}$ are exhibited as straight lines on the covering spaces of the tori.