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Séminaire Laurent Schwartz — EDP et applicationsTable of contents for this volume | Previous article | Next articleDidier Pilod Asymptotic Stability of Zakharov-Kuznetsov solitons Séminaire Laurent Schwartz — EDP et applications (2014-2015), Exp. No. 13, 12 p., doi: 10.5802/slsedp.73 Article PDF Résumé - Abstract In this report, we review the proof of the asymptotic stability of the Zakharov-Kuznetsov solitons in dimension two. Those results were recently obtained in a joint work with Raphaël Côte, Claudio Muñoz and Gideon Simpson. Bibliography [2] F. Béthuel, P. Gravejat and D. Smets, Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation, to appear in Ann. Sci. Éc. Norm. Supér., (2014) arXiv:1212.5027. [3] R. Côte, “Solitons et Dispersion”, Habilitation à Diriger des Recherches, Université de cergy-Pontoise, 2014. [4] R. Côte, C. Muñoz, D. Pilod and G. Simpson, Asymptotic stability of high-dimensional Zakharov-Kuznetsov solitons, preprint (2014), arXiv:1406.3196. [5] A. de Bouard, Stability and instability of some nonlinear dispersive solitary waves in higher dimension, Proc. Royal Soc. Edinburgh, 126 (1996), 89–112. MR 1378834 | Zbl 0861.35094 [6] K. El Dika, Asymptotic Stability of solitary waves for the Benjamin-Bona-Mahony equation, Disc. Cont. Dyn. Syst., 13 (2005), 583–622. MR 2152333 | Zbl 1083.35019 [7] A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equations 31 (1995), no. 6, 1002–1012. MR 1383936 | Zbl 0863.35097 [8] P. Gravejat and D. Smets, Asymptotic stability of the black soliton for the Gross-Pitaevskii equation, Proc. London Math. Soc. (2015) doi:10.1112/plms/pdv025. [9] A. Grünrock and S. Herr, The Fourier restriction norm method for the Zakharov-Kuznetsov equation, Disc. Contin. Dyn. Syst. Ser. A, 34 (2014), 2061–2068. MR 3124726 | Zbl 1280.35124 [10] D. Han-Kwan, From Vlasov-Poisson to Korteweg-de Vries and Zakharov-Kuznetsov, Comm. Math. Phys., 324 (2013), 961–993. MR 3123542 | Zbl 1284.35439 [11] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in Applied Mathematics, 93-128, Adv. Appl. Math. Suppl. Stud., 8, Academic Press, New York, 1983. MR 759907 | Zbl 0549.34001 [12] C. E. Kenig and Y. Martel, Asymptotic stability of solitons for the Benjamin-Ono equation, Rev. Mat. Iberoamericana, 25 (2009), no. 3, 909–970. MR 2590690 | Zbl 1247.35133 [13] E. A. Kuznetsov and V. E. Zakharov, On three dimensional solitons, Sov. Phys. JETP., 39 (1974), 285–286. [14] M. K. Kwong, Uniqueness of positive radial solutions of $\Delta u-u+u^p$ in $\mathbb{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243–266. MR 969899 | Zbl 0676.35032 [15] D. Lannes, F. Linares and J.-C. Saut, The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation, Prog. Nonlinear Diff. Eq. Appl., 84 (2013), 181–213. MR 3185896 | Zbl 1273.35263 [16] C. Laurent and Y. Martel, Smoothness and exponential decay of $L^2$-compact solutions of the generalized KdV equations, Comm. Part. Diff. Eq., 29 (2005), 157–171. MR 2038148 | Zbl 1140.35558 [17] F. Linares and A. Pastor, Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 41 (2009), no. 4, 1323–1339. MR 2540268 | Zbl 1197.35242 [18] Y. Martel, Linear Problems related to asymptotic stability of solitons of the generalized KdV equations, SIAM J. Math. Anal., 38 (2006), 759–781. MR 2262941 | Zbl 1126.35055 [19] Y. Martel and F. Merle, Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Ration. Mech. Anal., 157 (2001), 219–254. MR 1826966 | Zbl 0981.35073 [20] Y. Martel and F. Merle, Asymptotic Stability of solitons of the subcritical gKdV equations revisited, Nonlinearity, 18 (2005), 55–80. MR 2109467 | Zbl 1064.35171 [21] Y. Martel and F. Merle, Asymptotic stability of solitons of the gKdV equations with general nonlinearity, Math. Ann., 341 (2008), 391–427. MR 2385662 | Zbl 1153.35068 [22] Y. Martel, F. Merle, and T.P. Tsai, Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations, Comm. Math. Phys., 231 (2002) 347–373. MR 1946336 | Zbl 1017.35098 [23] L. Molinet and D. Pilod, Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications, Ann. Inst. H. Poincaré, Annal. Non., 32 (2015), 347–371. MR 3325241 [24] R. L. Pego and M. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305–349. MR 1289328 | Zbl 0805.35117 [25] F. Ribaud and S. Vento, Well-posedness results for the 3D Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 44 (2012), 2289–2304. MR 3023376 | Zbl 1251.35135 [26] M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472–491. MR 783974 | Zbl 0583.35028 |
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