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In this article, we study the half-wave map equation in dimension $1$ withimage on the sphere $S(\mathbb{R}^3)$. In particular, we focus on rationalsolutions and the well-posedness of the equation in this context. We prove theboundedness of the spins, global existence and non-turbulence of the solutionsunder different assumptions. Currently, only a very restrictive class ofsolutions have been studied.

Analytic proofs of well-posedness and non-turbulence results for the half-wave map equation