A Short Primer on the Half-Wave Maps Equation
Enno Lenzmann1
1 University of Basel Department of Mathematics Spiegelgasse 1, CH-4051 Basel Switzerland
Journées équations aux dérivées partielles (2018), Talk no. 4, 12 p.

We review the current state of results about the half-wave maps equation on the domain ℝ d with target 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.

Published online:
DOI: 10.5802/jedp.664
Enno Lenzmann 1

1 University of Basel Department of Mathematics Spiegelgasse 1, CH-4051 Basel Switzerland 
@incollection{JEDP_2018____A4_0,
     author = {Enno Lenzmann},
     title = {A {Short} {Primer} on the {Half-Wave} {Maps} {Equation}},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     note = {talk:4},
     pages = {1--12},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2018},
     doi = {10.5802/jedp.664},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.664/}
}
TY  - JOUR
AU  - Enno Lenzmann
TI  - A Short Primer on the Half-Wave Maps Equation
JO  - Journées équations aux dérivées partielles
N1  - talk:4
PY  - 2018
SP  - 1
EP  - 12
PB  - Groupement de recherche 2434 du CNRS
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.664/
DO  - 10.5802/jedp.664
LA  - en
ID  - JEDP_2018____A4_0
ER  - 
%0 Journal Article
%A Enno Lenzmann
%T A Short Primer on the Half-Wave Maps Equation
%J Journées équations aux dérivées partielles
%Z talk:4
%D 2018
%P 1-12
%I Groupement de recherche 2434 du CNRS
%U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.664/
%R 10.5802/jedp.664
%G en
%F JEDP_2018____A4_0
Enno Lenzmann. A Short Primer on the Half-Wave Maps Equation. Journées équations aux dérivées partielles (2018), Talk no. 4, 12 p. doi : 10.5802/jedp.664. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.664/

[1] Francesca Da Lio; Tristan Rivière Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps, Adv. Math., Volume 227 (2011) no. 3, pp. 1300-1348 | Zbl

[2] Patrick Gérard; Enno Lenzmann A Lax pair structure for the half-wave maps equation, Lett. Math. Phys., Volume 108 (2018) no. 7, pp. 1635-1648 | Zbl

[3] Patrick Gérard; Enno Lenzmann; Oana Pocovnicu; Pierre Raphaël A two-soliton with transient turbulent regime for the cubic half-wave equation on the real line, Ann. PDE, Volume 4 (2018) no. 1, 7, 166 pages | Zbl

[4] Joachim Krieger; Yannick Sire Small data global regularity for half-wave maps, Anal. PDE, Volume 11 (2018) no. 3, pp. 661-682 | Zbl

[5] Enno Lenzmann; Armin Schikorra On energy-critical half-wave maps into 𝕊 2 , Invent. Math., Volume 213 (2018) no. 1, pp. 1-82 | Zbl

[6] Vincent Millot; Yannick Sire On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into sphere, Arch. Ration. Mech. Anal., Volume 215 (2015) no. 1, pp. 125-210 | Zbl

[7] Vladimir V. Peller Hankel operators and their applications, Springer Monographs in Mathematics, Springer, 2003 | Zbl

[8] Xueke Pu; Boling Guo Well-posedness for the fractional Landau-Lifshitz equation without Gilbert damping, Calc. Var. Partial Differ. Equ., Volume 46 (2013) no. 3-4, pp. 441-460 | Zbl

[9] Yannick Sire; Juncheng Wei; Youquan Zheng Infinite time blow-up for half-harmonic map flow from ℝ into 𝕊 1 (2017) (https://arxiv.org/abs/1711.05387)

[10] Yannick Sire; Juncheng Wei; Youquan Zheng Nondegeneracy of half-harmonic maps from ℝ into 𝕊 1 , Proc. Am. Math. Soc., Volume 146 (2018) no. 12, pp. 5263-5268 | Zbl

[11] Tianci Zhou; Michael Stone Solitons in a continuous classical Haldane–Shastry spin chain, Phys. Lett., A, Volume 379 (2015) no. 43-44, pp. 2817-2825 | Zbl

Cited by Sources: