A Short Primer on the Half-Wave Maps Equation

Enno Lenzmann

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arXiv · 1903.01880

A Short Primer on the Half-Wave Maps Equation

Enno Lenzmann

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Abstract

We review the current state of results about the half-wave maps equation on the domain $\mathbb{R}^d$ with target $\mathbb{S}^2$. In particular, we focus on the energy-critical case $d=1$, where we discuss the classification of traveling solitary waves and a Lax pair structure together with its implications (e.\,g.~invariance of rational solutions and infinitely many conservation laws on a scale of homogeneous Besov spaces). Furthermore, we also comment on the one-dimensional space-periodic case. Finally, we list some open problem for future research.

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Enno Lenzmann. 2019-03-05. A Short Primer on the Half-Wave Maps Equation. https://arxiv.org/abs/1903.01880

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