On the asymptotic stability in the energy space for multi-solitons of the Landau-Lifshitz equation

Bahri, Yakine

doi:10.1090/tran/7108

On the asymptotic stability in the energy space for multi-solitons of the Landau-Lifshitz equation
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Trans. Amer. Math. Soc. 370 (2018), 4683-4707 Request permission

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