Asymptotic Stability of Zakharov-Kuznetsov solitons
Séminaire Laurent Schwartz — EDP et applications (2014-2015), Exposé no. 13, 12 p.

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.

@article{SLSEDP_2014-2015____A13_0,
author = {Pilod, Didier},
title = {Asymptotic Stability of Zakharov-Kuznetsov~solitons},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:13},
publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2014-2015},
doi = {10.5802/slsedp.73},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/slsedp.73/}
}
Pilod, Didier. Asymptotic Stability of Zakharov-Kuznetsov solitons. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Exposé no. 13, 12 p. doi : 10.5802/slsedp.73. http://archive.numdam.org/articles/10.5802/slsedp.73/

[1] H. Beresticky and P. L. Lions, Nonlinear scalar field equations, Arch. Rational Mech. Anal., 82 (1983), 313–345. | MR 695535 | Zbl 0533.35029

[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) . | Article

[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 ${ℝ}^{n}$, 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. | Numdam | 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