Geodesic
A Short Primer on the Half-Wave Maps Equation
Lenzmann, Enno1
1 University of Basel Department of Mathematics Spiegelgasse 1, CH-4051 Basel Switzerland
Journées équations aux dérivées partielles (2018), Exposé no. 4, 12 p.

Voir la notice de l'acte provenant de la source Numdam

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.

Publié le :
DOI : 10.5802/jedp.664

Lenzmann, Enno 1

1 University of Basel Department of Mathematics Spiegelgasse 1, CH-4051 Basel Switzerland 
@incollection{JEDP_2018____A4_0,
     author = {Lenzmann, Enno},
     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 = {http://geodesic.mathdoc.fr/articles/10.5802/jedp.664/}
}
TY  - JOUR
AU  - Lenzmann, Enno
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  - http://geodesic.mathdoc.fr/articles/10.5802/jedp.664/
DO  - 10.5802/jedp.664
LA  - en
ID  - JEDP_2018____A4_0
ER  - 
%0 Journal Article
%A Lenzmann, Enno
%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 http://geodesic.mathdoc.fr/articles/10.5802/jedp.664/
%R 10.5802/jedp.664
%G en
%F JEDP_2018____A4_0
Lenzmann, Enno. A Short Primer on the Half-Wave Maps Equation. Journées équations aux dérivées partielles (2018), Exposé no. 4, 12 p. doi : 10.5802/jedp.664. http://geodesic.mathdoc.fr/articles/10.5802/jedp.664/

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

[3] Gérard, Patrick; Lenzmann, Enno; Pocovnicu, Oana; Raphaël, Pierre 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] Krieger, Joachim; Sire, Yannick Small data global regularity for half-wave maps, Anal. PDE, Volume 11 (2018) no. 3, pp. 661-682 | Zbl

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

[6] Millot, Vincent; Sire, Yannick 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] Peller, Vladimir V. Hankel operators and their applications, Springer Monographs in Mathematics, Springer, 2003 | Zbl

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

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

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

Cité par Sources :