A Lax Pair Structure for the Half-Wave Maps Equation

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).

Included an explicit calculation of the Lax operator for a single speed soliton. Corrected some minor typos

References in corpus (2)

• On energy-critical half-wave maps into ${\mathbb{S}}^2$
• Semi-Classical Dynamics in Quantum Spin Systems

Cited by in corpus (10)

• An explicit formula for the Benjamin--Ono equation
• Multi-solitons of the half-wave maps equation and Calogero-Moser spin-pole dynamics
• Spin generalizations of the Benjamin-Ono equation
• The non-chiral intermediate Heisenberg ferromagnet equation
• Integrability, conservation laws and solitons of a many-body dynamical system associated with the half-wave maps equation
• Stationary states of the cubic conformal flow on ${\mathbb{S}}^3$
• On uniqueness for half-wave maps in dimension d >= 3
• Global Well-Posedness and Soliton Resolution for the Half-Wave Maps Equation with Rational Data
• Fifty Years of KdV: An Integrable System
• Periodic solutions of the non-chiral intermediate Heisenberg ferromagnet equation described by elliptic spin Calogero-Moser dynamics