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A Lax pair structure for the half-wave maps equation. (English) Zbl 1393.37074
Summary: 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 [E. Lenzmann and A. Schikorra, Invent. Math. 213, No. 1, 1–82 (2018; Zbl 1411.35208)], [T. Zhou and M. Stone, Phys. Lett., A 379, No. 43–44, 2817–2825 (2015; Zbl 1349.37076)] 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).

##### MSC:
 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
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