On the asymptotic stability in the energy space for multi-solitons of the Landau-Lifshitz equation
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- by Yakine Bahri PDF
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Abstract:
We establish the asymptotic stability of multi-solitons for the one-dimensional Landau-Lifshitz equation with an easy-plane anisotropy. The solitons have non-zero speed, are ordered according to their speeds and have sufficiently separated initial positions. We provide the asymptotic stability around solitons and between solitons. More precisely, we show that for an initial datum close to a sum of $N$ dark solitons, the corresponding solution converges weakly to one of the solitons in the sum, when it is translated to the center of this soliton, and converges weakly to zero when it is translated between solitons.References
- Yakine Bahri, Asymptotic stability in energy space for dark solitons of the Landau-Lifshitz equation, Anal. PDE 9 (2016), no. 3, 645–697. MR 3518533, DOI 10.2140/apde.2016.9.645
- I. Bejenaru, A. D. Ionescu, C. E. Kenig, and D. Tataru, Global Schrödinger maps in dimensions $d\geq 2$: small data in the critical Sobolev spaces, Ann. of Math. (2) 173 (2011), no. 3, 1443–1506. MR 2800718, DOI 10.4007/annals.2011.173.3.5
- Fabrice Béthuel, Philippe Gravejat, and Jean-Claude Saut, Existence and properties of travelling waves for the Gross-Pitaevskii equation, Stationary and time dependent Gross-Pitaevskii equations, Contemp. Math., vol. 473, Amer. Math. Soc., Providence, RI, 2008, pp. 55–103. MR 2522014, DOI 10.1090/conm/473/09224
- Fabrice Béthuel, Philippe Gravejat, and Didier Smets, Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation, Ann. Inst. Fourier (Grenoble) 64 (2014), no. 1, 19–70 (English, with English and French summaries). MR 3330540, DOI 10.5802/aif.2838
- Fabrice Béthuel, Philippe Gravejat, and Didier Smets, Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation, Ann. Sci. Éc. Norm. Supér. 48 (2015), no. 6, 1327–1381.
- S. Cuccagna and R. Jenkins, On asymptotic stability of $N$-solitons of the Gross-Pitaevskii equation, arXiv:1410.6887 (2014).
- André de Laire, Minimal energy for the traveling waves of the Landau-Lifshitz equation, SIAM J. Math. Anal. 46 (2014), no. 1, 96–132. MR 3148081, DOI 10.1137/130909081
- André de Laire and Philippe Gravejat, Stability in the energy space for chains of solitons of the Landau-Lifshitz equation, J. Differential Equations 258 (2015), no. 1, 1–80. MR 3271297, DOI 10.1016/j.jde.2014.09.003
- Ludwig D. Faddeev and Leon A. Takhtajan, Hamiltonian methods in the theory of solitons, Reprint of the 1987 English edition, Classics in Mathematics, Springer, Berlin, 2007. Translated from the 1986 Russian original by Alexey G. Reyman. MR 2348643
- Boling Guo and Shijin Ding, Landau-Lifshitz equations, Frontiers of Research with the Chinese Academy of Sciences, vol. 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. MR 2432099, DOI 10.1142/9789812778765
- A. Hubert and R. Schäfer, Magnetic domains: the analysis of magnetic microstructures, Springer-Verlag, Berlin-Heidelberg-New York, 1998.
- Robert L. Jerrard and Didier Smets, On Schrödinger maps from $T^1$ to $S^2$, Ann. Sci. Éc. Norm. Supér. (4) 45 (2012), no. 4, 637–680 (2013) (English, with English and French summaries). MR 3059243, DOI 10.24033/asens.2175
- A.M. Kosevich, B.A. Ivanov, and A.S. Kovalev, Magnetic solitons, Phys. Rep. 194 (1990), no. 3-4, 117–238.
- L.D. Landau and E.M. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Zeitsch. der Sow. 8 (1935), 153–169.
- Yvan Martel and Frank Merle, Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Ration. Mech. Anal. 157 (2001), no. 3, 219–254. MR 1826966, DOI 10.1007/s002050100138
- Yvan Martel and Frank Merle, Refined asymptotics around solitons for gKdV equations, Discrete Contin. Dyn. Syst. 20 (2008), no. 2, 177–218. MR 2358258, DOI 10.3934/dcds.2008.20.177
- Yvan Martel, Frank Merle, and Tai-Peng Tsai, Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations, Comm. Math. Phys. 231 (2002), no. 2, 347–373. MR 1946336, DOI 10.1007/s00220-002-0723-2
- A. V. Mikhaĭlov, The Landau-Lifschitz equation and the Riemann boundary problem on a torus, Phys. Lett. A 92 (1982), no. 2, 51–55. MR 677205, DOI 10.1016/0375-9601(82)90289-4
- Galina Perelman, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations, Comm. Partial Differential Equations 29 (2004), no. 7-8, 1051–1095. MR 2097576, DOI 10.1081/PDE-200033754
- Yu. L. Rodin, The Riemann boundary problem on Riemann surfaces and the inverse scattering problem for the Landau-Lifschitz equation, Phys. D 11 (1984), no. 1-2, 90–108. MR 762391, DOI 10.1016/0167-2789(84)90437-8
Additional Information
- Yakine Bahri
- Affiliation: Centre de Mathématiques Laurent Schwartz, École polytechnique, 91128 Palaiseau Cedex, France
- Address at time of publication: Department of Mathematics and Statistics, University of Victoria, 3800 Finnerty Road, Victoria, British Columbia V8P 5C2, Canada
- MR Author ID: 1170117
- Email: ybahri@uvic.ca
- Received by editor(s): April 25, 2016
- Received by editor(s) in revised form: September 21, 2016
- Published electronically: December 27, 2017
- Additional Notes: This work was supported by a Ph.D. grant from “Région Ile-de-France”
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 4683-4707
- MSC (2010): Primary 35B35, 35B40, 35Q51, 35C08, 35Q56; Secondary 35C07
- DOI: https://doi.org/10.1090/tran/7108
- MathSciNet review: 3812092