Asymptotic stability of solitons for nonlinear hyperbolic equations

E. A. Kopylova

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 68 (2013) no. 2, pp. 283-334

Résumé

Fundamental results on asymptotic stability of solitons are surveyed, methods for proving asymptotic stability are illustrated based on the example of a nonlinear relativistic wave equation with Ginzburg–Landau potential. Asymptotic stability means that a solution of the equation with initial data close to one of the solitons can be asymptotically represented for large values of the time as a sum of a (possibly different) soliton and a dispersive wave solving the corresponding linear equation. The proof techniques depend on the spectral properties of the linearized equation and may be regarded as a modern extension of the Lyapunov stability theory. Examples of nonlinear equations with prescribed spectral properties of the linearized dynamics are constructed. Bibliography: 45 titles.

Détail

Keywords: nonlinear hyperbolic equations, asymptotic stability, relativistic invariance, Hamiltonian structure, symplectic projection, invariant manifold, kink, Fermi's golden rule, scattering of solitons, asymptotic state.
Mots-clés : soliton

Bibliographie

[1] S. Agmon, “Spectral properties of Schrödinger operators and scattering theory”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2:2 (1975), 151–218 | MR | Zbl

[2] N. Boussaid, “Stable directions for small nonlinear Dirac standing waves”, Comm. Math. Phys., 268:3 (2006), 757–817 | DOI | MR | Zbl

[3] N. Boussaid, S. Cuccagna, “On stability of standing waves of nonlinear Dirac equations”, Comm. Partial Differential Equations, 37:6 (2012), 1001–1056 ; arXiv: 1103.4452v3 | DOI | MR | Zbl

[4] V. S. Buslaev, A. I. Komech, E. A. Kopylova, D. Stuart, “On asymptotic stability of solitary waves in Schrödinger equation coupled to nonlinear oscillator”, Comm. Partial Differential Equations, 33:4-6 (2008), 669–705 | DOI | MR | Zbl

[5] V. S. Buslaev, G. S. Perelman, “Scattering for the nonlinear Schrödinger equation: states close to a soliton”, St. Petersburg Math. J., 4:6 (1993), 1111–1142 | MR | Zbl

[6] V. S. Buslaev, G. S. Perelman, “On the stability of solitary waves for nonlinear Schrödinger equations”, Nonlinear evolution equations, Amer. Math. Soc. Transl. Ser. 2, 164, Amer. Math. Soc., Providence, RI, 1995, 75–98 | MR | Zbl

[7] V. S. Buslaev, C. Sulem, “On asymptotic stability of solitary waves for nonlinear Schrödinger equations”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20:3 (2003), 419–475 | DOI | MR | Zbl

[8] S. Cuccagna, “Stabilization of solutions to nonlinear Schrödinger equations”, Comm. Pure Appl. Math., 54:9 (2001), 1110–1145 | DOI | MR | Zbl

[9] S. Cuccagna, “On asymptotic stability in 3D of kinks for the $\phi^4$ model”, Trans. Amer. Math. Soc., 360:5 (2008), 2581–2614 | DOI | MR | Zbl

[10] W. Eckhaus, A. van Harten, The inverse scattering transformation and the theory of solitons. An introduction, North-Holland Math. Stud., 50, North-Holland Publishing Co., Amsterdam–New York, 1981, xi+222 pp. | MR | Zbl

[11] A. S. Fokas, V. E. Zakharov (eds.), Important developments in soliton theory, Springer Ser. Nonlinear Dynam., Springer-Verlag, Berlin, 1993, x+559 pp. | DOI | MR | Zbl

[12] C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura, “Method for solving the Korteweg–de Vries equation”, Phys. Rev. Lett., 19 (1967), 1095–1097 | DOI | Zbl

[13] W. Heisenberg, “Der derzeitige Stand der nichtlinearen Spinortheorie der Elementarteilchen”, Acta Phys. Austriaca, 14 (1961), 328–339 | MR | Zbl

[14] W. Heisenberg, Introduction to the unified field theory of elementary particles, Interscience Publishers, London–New York–Sydney, 1966, ix+177 pp. | Zbl

[15] V. Imaykin, A. Komech, H. Spohn, “Scattering asymptotics for a charged particle coupled to the Maxwell field”, J. Math. Phys., 52:4 (2011), 042701, 33 pp. | DOI | MR

[16] V. Imaikin, A. Komech, B. Vainberg, “On scattering of solitons for the Klein–Gordon equation coupled to a particle”, Comm. Math. Phys., 268:2 (2006), 321–367 | DOI | MR | Zbl

[17] E. Kirr, A. Zarnesku, “On the asymptotic stability of bound states in 2D cubic Schrödinger equation”, Comm. Math. Phys., 272:2 (2007), 443–468 | DOI | MR | Zbl

[18] A. Komech, E. Kopylova, “Scattering of solitons for Schrödinger equation coupled to a particle”, Russ. J. Math. Phys., 13:2 (2006), 158–187 ; arXiv: math/0609649 | DOI | MR | Zbl

[19] A. I. Komech, E. A. Kopylova, “Weighted energy decay for 1D Klein–Gordon equation”, Comm. Partial Differential Equations, 35:2 (2010), 353–374 | DOI | MR | Zbl

[20] A. I. Komech, E. A. Kopylova, H. Spohn, “Scattering of solitons for Dirac equation coupled to a particle”, J. Math. Anal. Appl., 383:2 (2011), 265–290 ; arXiv: 1012.3109 | DOI | MR | Zbl

[21] A. Komech, E. Kopylova, D. Stuart, “On asymptotic stability of solitons in a nonlinear Schrödinger equation”, Commun. Pure Appl. Anal., 11:3 (2012), 1063–1079 ; arXiv: 0807.1878 | DOI | MR

[22] A. I. Komech, N. J. Mauser, A. P. Vinnichenko, “Attraction to solitons in relativistic nonlinear wave equations”, Russ. J. Math. Phys., 11:3 (2004), 289–307 | MR | Zbl

[23] E. Kopylova, “On the asymptotic stability of solitary waves in the discrete Schrödinger equation coupled to a nonlinear oscillator”, Nonlinear Anal., 71:7-8 (2009), 3031–3046 ; arXiv: 0805.3403 | DOI | MR | Zbl

[24] E. A. Kopylova, “Dispersive estimates for the Schrödinger and Klein–Gordon equation”, Russian Math. Surveys, 65:1 (2010), 95–142 | DOI | DOI | MR | Zbl

[25] E. A. Kopylova, “On asymptotic stability of solitary waves in discrete Klein–Gordon equation coupled to nonlinear oscillator”, Appl. Anal., 89:9 (2010), 1467–1492 | DOI | MR | Zbl

[26] E. Kopylova, “On long-time decay for modified Klein–Gordon equation”, Commun. Math. Anal., Conference 03 (2011), 137–152 ; arXiv: 1009.2649 | MR | Zbl

[27] E. A. Kopylova, A. I. Komech, “On asymptotic stability of moving kink for relativistic Ginzburg–Landau equation”, Comm. Math. Phys., 302:1 (2011), 225–252 ; arXiv: 0910.5538 | DOI | MR | Zbl

[28] E. Kopylova, A. I. Komech, “On asymptotic stability of kink for relativistic Ginzburg–Landau equation”, Arch. Ration. Mech. Anal., 202:1 (2011), 213–245 ; arXiv: 0910.5539 | DOI | MR | Zbl

[29] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris; Gauthier-Villars, Paris, 1969, xx+554 pp. | MR | MR | Zbl | Zbl

[30] J. R. Miller, M. I. Weinstein, “Asymptotic stability of solitary waves for the regularized long-wave equation”, Comm. Pure Appl. Math., 49:4 (1996), 399–441 | 3.0.CO;2-7 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[31] M. Murata, “Asymptotic expansions in time for solutions of Schrödinger-type equations”, J. Funct. Anal., 49:1 (1982), 10–56 | DOI | MR | Zbl

[32] R. L. Pego, M. I. Weinstein, “Asymptotic stability of solitary waves”, Comm. Math. Phys., 164:2 (1994), 305–349 | DOI | MR | Zbl

[33] D. E. Pelinovsky, A. A. Stefanov, “Asymptotic stability of small gap solitons in nonlinear Dirac equations”, J. Math. Phys., 53:7 (2012), 073705, 27 pp. ; arXiv: 1008.4514 | DOI | MR

[34] C.-A. Pillet, C. E. Wayne, “Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations”, J. Differential Equations, 141:2 (1997), 310–326 | DOI | MR | Zbl

[35] M. Reed, Abstract non-linear wave equations, Lecture Notes in Math., 507, Springer-Verlag, Berlin–New York, 1976, vi+128 pp. | DOI | MR | Zbl

[36] M. Reed, B. Simon, Methods of modern mathematical physics, v. III, Scattering theory, Academic Press, New York–London, 1979, xv+463 pp. | MR | MR | Zbl | Zbl

[37] I. Rodnianski, W. Schlag, A. Soffer, “Dispersive analysis of charge transfer models”, Comm. Pure Appl. Math., 58:2 (2005), 149–216 | DOI | MR | Zbl

[38] A. Soffer, M. I. Weinstein, “Multichannel nonlinear scattering for nonintegrable equations”, Comm. Math. Phys., 133:1 (1990), 119–146 | DOI | MR | Zbl

[39] A. Soffer, M. I. Weinstein, “Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data”, J. Differential Equations, 98:2 (1992), 376–390 | DOI | MR | Zbl

[40] A. Soffer, M. I. Weinstein, “Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations”, Invent. Math., 136:1 (1999), 9–74 | DOI | MR | Zbl

[41] W. A. Strauss, “Nonlinear invariant wave equations”, Invariant wave equations, Proceedings of the “Ettore Majorana” International School of Mathematical Physics (Erice, 1977), Lecture Notes in Phys., 73, Springer, Berlin–New York, 1978, 197–249 | DOI | MR | Zbl

[42] Tai-Peng Tsai, Horng-Tzer Yau, “Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions”, Comm. Pure Appl. Math., 55:2 (2002), 153–216 | DOI | MR | Zbl

[43] Tai-Peng Tsai, “Asymptotic dynamics of nonlinear Schrödinger equations with many bound states”, J. Differential Equations, 192:1 (2003), 225–282 | DOI | MR | Zbl

[44] M. Weinstein, “Modulational stability of ground states of nonlinear Schrödinger equations”, SIAM J. Math. Anal., 16:3 (1985), 472–491 | DOI | MR | Zbl

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