LAUSR

AbstractWe consider the half-wave maps equation $$\begin{aligned} \partial _t \vec {S} = \vec {S} \wedge |\nabla | \vec {S}, \end{aligned}$$∂tS→=S→∧|∇|S→,where $$\vec {S}= \vec {S}(t,x)$$S→=S→(t,x) takes values on the two-dimensional unit sphere $$\mathbb {S}^2$$S2 and $$x \in \mathbb {R}$$x∈R (real line case) or $$x \in \mathbb {T}$$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 $$\mathbb {H}^2$$H2 (hyperbolic plane).