We consider the half-wave maps equation $$ \partial_t \vec{ S} = \vec{ S} \wedge |\nabla| \vec{ S}, $$ where $\vec{ S}= \vec{ S}(t,x)$ takes values on the two-dimensional unit sphere $\mathbb{ S}^2$ and $x \in \mathbb{ R}$ (real line case) or $x \in \mathbb{ T}$ (periodic case). This an energy-critical Hamiltonian evolution equation recently introduced in \cite{ LS,Zh}, 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 $\mathbb{ H}^2$ (hyperbolic plane).
Categories: math.AP math-ph math.MP