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

2024/12/04 by Gérard, Patrick, Lenzmann, Enno · 3 citations
#Analysis of PDEs (math.AP) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2412.03351

Abstract

We study the energy-critical half-wave maps equation: ∂_t u = u × |D| u for u : [0, T) × ℝ → \mathbbS². Our main result establishes the global existence and uniqueness of solutions for all rational initial data u₀ : ℝ → \mathbbS². This demonstrates global well-posedness for a dense subset within the scaling-critical energy space H^1/2(ℝ; \mathbbS²). Furthermore, we prove soliton resolution for a dense subset of initial data in the energy space, with uniform bounds for all higher Sobolev norms 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 Gr_k(ℂ^d). Notably, this includes the complex projective spaces \mathbbCP^d-1 ≅ Gr₁(ℂ^d) thereby extending the classical case of the target \mathbbS² ≅ \mathbbCP¹.

Cited by

• Scattering of the defocusing Calogero--Moser derivative nonlinear Schrödinger equation
• A superintegrable quantum field theory
• A log-log upper bound on blow-up rates for the mass-critical half-wave equation

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