On asymptotic stability of moving kink for relativistic Ginzburg-Landau equation. (English) Zbl 1209.35134

Summary: We prove the asymptotic stability of the moving kinks for the nonlinear relativistic wave equations in one space dimension with a Ginzburg-Landau potential: starting in a small neighborhood of the kink, the solution, asymptotically in time, is the sum of a uniformly moving kink and dispersive part described by the free Klein-Gordon equation. The remainder decays in a global energy norm. Our recent results on the weighted energy decay for the Klein-Gordon equations play a crucial role in the proofs.


35Q56 Ginzburg-Landau equations
35Q75 PDEs in connection with relativity and gravitational theory
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35C08 Soliton solutions
Full Text: DOI arXiv


[1] Agmon S.: Spectral properties of Schrödinger operator and scattering theory. Ann. Scuola Norm. Sup. Pisa, Ser. IV 2, 151–218 (1975) · Zbl 0315.47007
[2] Bais, F.A.: Topological excitations in gauge theories; An introduction from the physical point of view. Springer Lecture Notes in Mathematics, Vol. 926, Berlin-Heidelberg-New York: Springer, 1982 · Zbl 0487.58033
[3] Bjørn F.: Geometry, Particles, and Fields. Springer, NY, New York (1998) · Zbl 0897.53001
[4] Buslaev V.S., Perelman G.S.: Scattering for the nonlinear Schrödinger equations: states close to a soliton. St. Petersburg Math. J. 4(6), 1111–1142 (1993)
[5] Buslaev V.S., Sulem C.: On asymptotic stability of solitary waves for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré. Anal. Non Linéaire 20(3), 419–475 (2003) · Zbl 1028.35139
[6] Cuccagna S.: Stabilization of solutions to nonlinear Schrödinger equations. Comm. Pure Appl. Math. 54, 1110–1145 (2001) · Zbl 1031.35129
[7] Cuccagna S.: On asymptotic stability in 3D of kinks for the \({\phi^4}\) model. Transactions of AMS 360(5), 2581–2614 (2008) · Zbl 1138.35062
[8] Jensen A., Kato T.: Spectral properties of Schrödinger operators and time-decay of the wave functions. Duke Math. J. 46, 583–611 (1979) · Zbl 0448.35080
[9] Jensen A., Nenciu G.: A unified approach to resolvent expansions at thresholds. Rev. Math. Phys. 13(6), 717–754 (2001) · Zbl 1029.81067
[10] Henry D.B., Perez J.F., Wreszinski W.F.: Stability theory for solitary-wave solutions of scalar field equations. Commun. Math. Phys. 85, 351–361 (1982) · Zbl 0546.35062
[11] Imaikin V., Komech A.I., Vainberg B.: On scattering of solitons for the Klein-Gordon equation coupled to a particle. Commun. Math. Phys. 268(2), 321–367 (2006) · Zbl 1127.35054
[12] Kirr E., Zarnesku A.: On the asymptotic stability of bound states in 2D cubic Schrödinger equation. Commun. Math. Phys. 272(2), 443–468 (2007) · Zbl 1194.35416
[13] Komech A., Kopylova E.: Weighted energy decay for 1D Klein-Gordon equation. Comm. PDE 35(2), 353–374 (2010) · Zbl 1190.35134
[14] Kopylova, E.: On long-time decay for Klein-Gordon equation. Comm. Math. Anal. Conference 03, 137–152 (2011). http://arriv.org/abs/1009.2649vz [math-ph]; 2010 · Zbl 1213.35106
[15] Lions, J.L.: Quelques Mèthodes de Rèsolution des Problémes aux Limites non Linéaires. Paris: Dunod, 1969 · Zbl 0189.40603
[16] Murata M.: Asymptotic expansions in time for solutions of Schrödinger-type equations. J. Funct. Anal. 49, 10–56 (1982) · Zbl 0499.35019
[17] Miller J., Weinstein M.: Asymptotic stability of solitary waves for the regularized long-wave equation Comm. Pure Appl. Math. 49(4), 399–441 (1996) · Zbl 0854.35102
[18] Pego R.L., Weinstein M.I.: Asymptotic stability of solitary waves, Commun. Math. Phys. 164, 305–349 (1994) · Zbl 0805.35117
[19] Pillet C.A., Wayne C.E.: Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations. J. Differ. Eq. 141(2), 310–326 (1997) · Zbl 0890.35016
[20] Reed M.: Abstract Non-Linear Wave Equations Lecture Notes in Mathematics 507. Springer, Berlin (1976)
[21] Reed M., Simon B.: Methods of Modern Mathematical Physics, III. Academic Press, New York (1979) · Zbl 0405.47007
[22] Rodnianski I., Schlag W., Soffer A.: Dispersive analysis of charge transfer models. Commun. Pure Appl. Math. 58(2), 149–216 (2005) · Zbl 1130.81053
[23] Soffer A., Weinstein M.I.: Multichannel nonlinear scattering for nonintegrable equations. Commun. Math. Phys. 133, 119–146 (1990) · Zbl 0721.35082
[24] Soffer A., Weinstein M.I.: Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data. J. Diff. Eq. 98(2), 376–390 (1992) · Zbl 0795.35073
[25] Soffer A., Weinstein M.I.: Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 136, 9–74 (1999) · Zbl 0910.35107
[26] Strauss W.A.: Nonlinear invariant wave equations Lecture Notes in Physics 73, pp. 197–249. Springer, Berlin (1978)
[27] Tsai T.-P., Yau H.-T.: Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions. Commun. Pure Appl. Math. 55(2), 153–216 (2002) · Zbl 1031.35137
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.