A Lax Pair Structure for the Half-Wave Maps Equation

We consider the half-wave maps equation
∂tS⃗ =S⃗ ∧|∇|S⃗ ,
where S⃗ =S⃗ (t,x) takes values on the two-dimensional unit sphere S2 and x∈R (real line case) or x∈T (periodic case). This an energy-critical Hamiltonian evolution equation recently introduced in Lenzmann and Schikorra (2017, arXiv:1702.05995v2), Zhou and Stone (Phys Lett A 379:2817–2825, 2015) which formally arises as an effective evolution equation in the classical and continuum limit of Haldane–Shastry quantum spin chains. We prove that the half-wave maps equation admits a Lax pair and we discuss some analytic consequences of this finding. As a variant of our arguments, we also obtain a Lax pair for the half-wave maps equation with target H2 (hyperbolic plane).