Vol. 9, No. 3, 2016

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
Editors' interests
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author index
To appear
Other MSP journals
Asymptotic stability in energy space for dark solitons of the Landau–Lifshitz equation

Yakine Bahri

Vol. 9 (2016), No. 3, 645–697

We prove the asymptotic stability in energy space of nonzero speed solitons for the one-dimensional Landau–Lifshitz equation with an easy-plane anisotropy

tm + m × (xxm m3e3) = 0

for a map m = (m1,m2,m3) : × S2, where e3 = (0,0,1). More precisely, we show that any solution corresponding to an initial datum close to a soliton with nonzero speed is weakly convergent in energy space as time goes to infinity to a soliton with a possible different nonzero speed, up to the invariances of the equation. Our analysis relies on the ideas developed by Martel and Merle for the generalized Korteweg–de Vries equations. We use the Madelung transform to study the problem in the hydrodynamical framework. In this framework, we rely on the orbital stability of the solitons and the weak continuity of the flow in order to construct a limit profile. We next derive a monotonicity formula for the momentum, which gives the localization of the limit profile. Its smoothness and exponential decay then follow from a smoothing result for the localized solutions of the Schrödinger equations. Finally, we prove a Liouville type theorem, which shows that only the solitons enjoy these properties in their neighbourhoods.

asymptotic stability, solitons, Landau–Lifshitz equation, travelling waves
Mathematical Subject Classification 2010
Primary: 35Q51, 35Q60, 37K40
Received: 16 July 2015
Revised: 25 November 2015
Accepted: 30 January 2016
Published: 17 June 2016
Yakine Bahri
Centre de Mathématiques Laurent Schwartz
École Polytechnique
91128 Palaiseau Cedex