Global Well-Posedness and Soliton Resolution for the Half-Wave Maps Equation with Rational Data

We study the energy-critical half-wave maps equation: $${\mathrm{\partial }}_t\mathbf{u}=\mathbf{u}\times \mathrm{\mid }D\mathrm{\mid }\mathbf{u}$$ for $\mathbf{u}:[0,T)\times \mathbb{R}\to {\mathbb{S}}^2$. Our main result establishes the global existence and uniqueness of solutions for all rational initial data ${\mathbf{u}}_0:\mathbb{R}\to {\mathbb{S}}^2$. This demonstrates global well-posedness for a dense subset within the scaling-critical energy space ${\dot{H}}^{1\mathrm{/}2}(\mathbb{R};{\mathbb{S}}^2)$. Furthermore, we prove soliton resolution for a dense subset of initial data in the energy space, with uniform bounds for all higher Sobolev norms ${\dot{H}}^s$ for $s>0$. Our analysis utilizes the Lax pair structure of the half-wave maps equation on Hardy spaces in combination with an explicit flow formula. Extending these results, we establish global well-posedness for rational initial data (along with a soliton resolution result) for a generalized class of matrix-valued half-wave maps equations with target spaces in the complex Grassmannians ${\mathbf{G}\mathbf{r}}_k({\mathbb{C}}^d)$. Notably, this includes the complex projective spaces ${\mathbb{C}\mathbb{P}}^{d-1}\cong {\mathbf{G}\mathbf{r}}_1({\mathbb{C}}^d)$ thereby extending the classical case of the target ${\mathbb{S}}^2\cong {\mathbb{C}\mathbb{P}}^1$.