We describe some recent results concerning the nonlinear ${L}^{2}$-stability of multi-solitons of the Korteweg-de Vries equation [4], and ${H}^{1}$-stability of multi-kinks of the modified Korteweg-de Vries [49]. The proof of both results is closely linked to stability properties for solitons of the integrable Gardner equation, which have been considered by Martel, Merle and Tsai [41, 40].

Keywords: KdV equation, modified KdV equation, Gardner equation, integrability, multi-soliton, multi-kink, stability, asymptotic stability, Gardner transform

@article{JEDP_2011____A8_0, author = {Mu\~noz, Claudio}, title = {$H^1$-stability of mKdV multi-kinks}, journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, publisher = {Groupement de recherche 2434 du CNRS}, year = {2011}, doi = {10.5802/jedp.80}, language = {en}, url = {http://www.numdam.org/item/JEDP_2011____A8_0} }

Muñoz, Claudio. $H^1$-stability of mKdV multi-kinks. Journées équations aux dérivées partielles, (2011), article no. 8, 16 p. doi : 10.5802/jedp.80. http://www.numdam.org/item/JEDP_2011____A8_0/

[1] M. Ablowitz, and P. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series, 149. Cambridge University Press, Cambridge, 1991. | MR 1149378 | Zbl 0762.35001

[2] M. Ablowitz, D. Kaup, A. Newell, and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Studies in Appl. Math. 53 (1974), no. 4, 249–315. | MR 450815 | Zbl 0408.35068

[3] M. Ablowitz, and H. Segur, Solitons and the inverse scattering transform, SIAM Studies in Applied Mathematics, 4. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981. x+425 pp. | MR 642018 | Zbl 0472.35002

[4] M. A. Alejo, C. Muñoz, and L. Vega, The Gardner equation and the ${L}^{2}$-stability of the $N$-soliton solutions of the Korteweg-de Vries equation, to appear in Transactions of the AMS.

[5] T.B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. London A **328**, (1972) 153–183.
| MR 338584

[6] H. Berestycki, and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rat. Mech. Anal. **82** (1983), 313–345.
| MR 695535
| Zbl 0533.35029

[7] F. Béthuel, P. Gravejat, J.-C. Saut, and D. Smets, Orbital stability of the black soliton to the Gross-Pitaevskii equation, Indiana Univ. Math. J. **57** (2008), no. 6, 2611–2642. .
| MR 2482993
| Zbl 1171.35012

[8] J.L. Bona, P. Souganidis and W. Strauss, Stability and instability of solitary waves of Korteweg-de Vries type, Proc. Roy. Soc. London **411** (1987), 395–412.
| MR 897729
| Zbl 0648.76005

[9] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV equation, Geom. Funct. Anal. **3** (1993), no. 3, 209-262.
| MR 1215780
| Zbl 0787.35098

[10] K.W. Chow, R.H.J Grimshaw, and E. Ding, Interactions of breathers and solitons in the extended Korteweg-de Vries equation, Wave Motion **43** (2005) 158–166.
| MR 2186925

[11] M. Christ, J. Colliander, and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math. **125** (2003), no. 6, 1235–1293.
| MR 2018661
| Zbl 1048.35101

[12] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T.Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$, J. Amer. Math. Soc. **16** (2003), no. 3, 705–749 (electronic).
| MR 1969209
| Zbl 1025.35025

[13] S. Cuccagna, On asymptotic stability in 3D of kinks for the ${\phi}^{4}$ model, Trans. Amer. Math. Soc. **360** (2008), no. 5, 2581–2614.
| MR 2373326
| Zbl 1138.35062

[14] T. Dauxois, and M. Peyrard, Physics of solitons, Cambridge University Press, 2006. | Zbl 1192.35001

[15] E. Fermi, J. Pasta and S. Ulam, Studies of nonlinear problems I, Los Alamos Report LA1940 (1955); reproduced in Nonlinear Wave Motion, A.C. Newell, ed., Am. Math. Soc., Providence, R. I., 1974, pp. 143–156. | MR 336014 | Zbl 0353.70028

[16] C.S. Gardner, M.D. Kruskal, and R. Miura, Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion, J. Math. Phys. **9**, no. 8 (1968), 1204–1209.
| MR 252826
| Zbl 0283.35019

[17] P. Gérard, and Z. Zhang, Orbital stability of traveling waves for the one-dimensional Gross-Pitaevskii equation, J. Math. Pures Appl. (9) **91** (2009), no. 2, 178–210.
| MR 2498754
| Zbl pre05532849

[18] F. Gesztesy, and B. Simon, Constructing solutions of the mKdV-equation, J. Funct. Anal. **89** (1990), no. 1, 53–60.
| MR 1040955
| Zbl 0711.35121

[19] F. Gesztesy, W. Schweiger, and B. Simon, Commutation methods applied to the mKdV-equation, Trans. AMS **324** (1991), no. 2, 465–525.
| MR 1029000
| Zbl 0728.35106

[20] M. Grillakis, J. Shatah, and W. Strauss, Stability theory of solitary waves in the presence of symmetry, I, J. Funct. Anal. **74** (1987), 160–197.
| MR 901236
| Zbl 0656.35122

[21] H. Grosse, Solitons of the modified KdV equation, Lett. Math. Phys. **8** (1984), 313-319.
| MR 759630
| Zbl 0557.35117

[22] H. Grosse, New solitons connected to the Dirac equation, Phys. Rep. **134** (1986), 297–304.
| MR 832137

[23] R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., **27** (1971), 1192–1194.
| Zbl 1168.35423

[24] D. B. Henry, J.F. Perez; and W. F. Wreszinski, Stability Theory for Solitary-Wave Solutions of Scalar Field Equations, Comm. Math. Phys. **85**, 351–361(1982).
| MR 678151
| Zbl 0546.35062

[25] T. Kappeler, and P. Topalov, Global fold structure of the Miura map on ${L}^{2}\left(\mathbb{T}\right)$, Int. Math. Res. Not. **2004**, no. 39, 2039–2068.
| MR 2062735
| Zbl 1076.35111

[26] C.E. Kenig, and Y. Martel, Global well-posedness in the energy space for a modified KP II equation via the Miura transform, Trans. Amer. Math. Soc. **358** no. 6, pp. 2447–2488.
| MR 2204040
| Zbl 1106.35082

[27] C.E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle, Comm. Pure Appl. Math. **46**, (1993) 527–620.
| MR 1211741
| Zbl 0808.35128

[28] C.E. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. **106** (2001), no. 3, 617–633.
| MR 1813239
| Zbl 1034.35145

[29] E. Kopylova, and A. I. Komech, On Asymptotic Stability of Kink for Relativistic Ginzburg-Landau Equations, to appear in Arch. Rat. Mech. Anal. | MR 2835867

[30] D.J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of stationary waves, Philos. Mag. Ser. 5, **39** (1895), 422–443.

[31] M.D. Kruskal and N.J. Zabusky, Interaction of “solitons” in a collisionless plasma and recurrence of initial states, Phys. Rev. Lett. **15** (1965), 240–243.
| Zbl 1201.35174

[32] P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. **21**, (1968) 467–490.
| MR 235310
| Zbl 0162.41103

[33] J.H. Maddocks, and R.L. Sachs, On the stability of KdV multi-solitons, Comm. Pure Appl. Math. **46**, 867–901 (1993).
| MR 1220540
| Zbl 0795.35107

[34] Y. Martel, Asymptotic $N$-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations, Amer. J. Math. **127** (2005), no. 5, 1103–1140.
| MR 2170139
| Zbl 1090.35158

[35] Y. Martel, and F. Merle, Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Ration. Mech. Anal. **157** (2001), no. 3, 219–254.
| MR 1826966
| Zbl 0981.35073

[36] 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

[37] Y. Martel and F. Merle, Description of two soliton collision for the quartic gKdV equation, preprint arXiv:0709.2672 (2007), to appear in Annals of Mathematics. | MR 2831108

[38] Y. Martel and F. Merle, Stability of two soliton collision for nonintegrable gKdV equations, Comm. Math. Phys. **286** (2009), 39–79.
| MR 2470923
| Zbl 1179.35291

[39] Y. Martel and F. Merle, Inelastic interaction of nearly equal solitons for the quartic gKdV equation, to appear in Inventiones Mathematicae. | MR 2772088 | Zbl pre05876207

[40] Y. Martel, and F. Merle, Asymptotic stability of solitons of the gKdV equations with general nonlinearity, Math. Ann. **341** (2008), no. 2, 391–427.
| MR 2385662
| Zbl 1153.35068

[41] 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

[42] F. Merle; and L. Vega, ${L}^{2}$ stability of solitons for KdV equation, Int. Math. Res. Not. 2003, no. 13, 735–753. | MR 1949297 | Zbl 1022.35061

[43] R.M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Math. Phys. **9**, no. 8 (1968), 1202–1204.
| MR 252825
| Zbl 0283.35018

[44] R.M. Miura, The Korteweg–de Vries equation: a survey of results, SIAM Review **18**, (1976) 412–459.
| MR 404890
| Zbl 0333.35021

[45] T. Mizumachi, and D. Pelinovsky, Bäcklund transformation and ${L}^{2}$-stability of NLS solitons, preprint.

[46] T. Mizumachi, and N. Tzvetkov, Stability of the line soliton of the KP–II equation under periodic transverse perturbations, preprint. | MR 2885592

[47] C. Muñoz, On the inelastic 2-soliton collision for gKdV equations with general nonlinearity, Int. Math. Research Notices (2010) **2010** (9): 1624–1719.
| MR 2643578
| Zbl 1198.35234

[48] C. Muñoz, ${L}^{2}$-stability of multi-solitons, Séminaire EDP et Applications, École Polythecnique, France, Janvier 2011 http://www.dim.uchile.cl/~cmunoz.

[49] C. Muñoz, The Gardner equation and the stability of multi-kink solutions of the mKdV equation, preprint arXiv:1106.0648.

[50] R.L. Pego, and M.I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys. **164**, 305–349 (1994).
| MR 1289328
| Zbl 0805.35117

[51] Soffer, A.; Weinstein, M. I. Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math. **136** (1999), no. 1, 9–74.
| MR 1681113
| Zbl 0910.35107

[52] B. Thaller, The Dirac equation, Texts and Monographs in Physics. Springer-Verlag, Berlin, 1992. xviii+357 pp. | MR 1219537 | Zbl 0765.47023

[53] M.I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure. Appl. Math. **39**, (1986) 51—68.
| MR 820338
| Zbl 0594.35005

[54] M.I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. **16** (1985), no. 3, 472–491.
| MR 783974
| Zbl 0583.35028

[55] M.V. Wickerhauser, Inverse scattering for the heat operator and evolutions in $2+1$ variables, Comm. Math. Phys. **108** (1987), 67–89.
| MR 872141
| Zbl 0633.35070

[56] P. E. Zhidkov, Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Mathematics, vol. 1756, Springer-Verlag, Berlin, 2001. | MR 1831831 | Zbl 0987.35001