$H^1$-stability of mKdV multi-kinks

Muñoz, Claudio

Journées équations aux dérivées partielles (2011), article no. 8, 16 p.

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].

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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

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

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

[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 | Zbl

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

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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

[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

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

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

[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 | Zbl

[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 | Zbl

[36] Y. Martel and F. Merle, Asymptotic stability of solitons of the subcritical gKdV equations revisited, Nonlinearity 18 (2005) 55–80. | MR | Zbl

[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

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

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

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