A Lax Pair Structure for the Half-Wave Maps Equation

TL;DR

This paper establishes a Lax pair structure for the energy-critical half-wave maps equation, revealing integrability properties and extending the framework to hyperbolic target spaces, with implications for quantum spin chain models.

Contribution

It introduces a Lax pair for the half-wave maps equation, demonstrating its integrability and extending the results to hyperbolic target spaces.

Findings

Proves the existence of a Lax pair for the half-wave maps equation.

Discusses analytic consequences of the Lax pair structure.

Extends the Lax pair construction to hyperbolic target spaces.

Abstract

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).