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On asymptotic stability of kink for relativistic Ginzburg-Landau equations. (English) Zbl 1256.35146

Summary: We prove the asymptotic stability of kink for the nonlinear relativistic wave equations of the Ginzburg-Landau type in one space dimension: for any odd initial condition in a small neighborhood of the kink, the solution, asymptotically in time, is the sum of the kink and dispersive part described by the free Klein-Gordon equation. The remainder converges to zero in a global norm.

MSC:

35Q56 Ginzburg-Landau equations
35Q75 PDEs in connection with relativity and gravitational theory
83A05 Special relativity
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[1] 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)
[2] Buslaev V.S., Perelman G.S.: On the stability of solitary waves for nonlinear Schrödinger equations. Am. Math. Soc. Trans. 164(2), 75-98 (1995) · Zbl 0841.35108
[3] 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 · doi:10.1016/S0294-1449(02)00018-5
[4] Cuccagna S.: Stabilization of solutions to nonlinear Schrödinger equations. Commun. Pure Appl. Math. 54, 1110-1145 (2001) · Zbl 1031.35129 · doi:10.1002/cpa.1018
[5] Cuccagna S.: On asymptotic stability of ground states of NLS. Rev. Math. Phys. 15, 877-903 (2003) · Zbl 1084.35089 · doi:10.1142/S0129055X03001849
[6] Cuccagna S., Mizumachi T.: On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations. Commun. Math. Phys. 284(1), 51-77 (2008) · Zbl 1155.35092 · doi:10.1007/s00220-008-0605-3
[7] Cuccagna S.: On asymptotic stability in 3D of kinks for the \[{\phi^4}\] model. Trans. AMS 360(5), 2581-2614 (2008) · Zbl 1138.35062 · doi:10.1090/S0002-9947-07-04356-5
[8] Deift, P.A., Its, A.R., Zhou, X.: Long-time asymptotics for integrable nonlinear wave equations. Important Developments in Soliton Theory (Eds. Fokas A.S. and Zakharov V.E.). Springer, Berlin, 181-204, 1993 · Zbl 0926.35132
[9] Faddeev L.D., Takhtadzhyan L.A.: Hamiltonian Methods in the Theory of Solitons. Springer, Berlin (1987) · Zbl 0632.58004 · doi:10.1007/978-3-540-69969-9
[10] Gol’dman I.I., Krivchenkov V.D., Kogan V.I., Galitskii V.M.: Problems in Quantum Mechanics. Infosearch LTD., London (1960)
[11] 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 · doi:10.1007/BF01208719
[12] 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 · doi:10.1007/s00220-006-0088-z
[13] 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 · doi:10.1007/s00220-007-0233-3
[14] Komech A., Kopylova E.: Weighted energy decay for 1D Klein-Gordon equation. Commun. PDE 35(2), 353-374 (2010) · Zbl 1190.35134 · doi:10.1080/03605300903419783
[15] Kopylov, S.A.: Private communication
[16] Lions, J.L.: Quelques Mèthodes de Rèsolution des Problémes aux Limites non Linéaires. Paris, Dunod, 1969 · Zbl 0189.40603
[17] Merkli M., Sigal I.M.: A time-dependent theory of quantum resonances. Commun. Math. Phys. 201(3), 549-576 (1999) · Zbl 0934.47007 · doi:10.1007/s002200050568
[18] Miller J., Weinstein M.: Asymptotic stability of solitary waves for the regularized long-wave equation. Commun. Pure Appl. Math. 49(4), 399-441 (1996) · Zbl 0854.35102 · doi:10.1002/(SICI)1097-0312(199604)49:4<399::AID-CPA4>3.0.CO;2-7
[19] Pego R.L., Weinstein M.I.: Asymptotic stability of solitary waves. Commun. Math. Phys. 164, 305-349 (1994) · Zbl 0805.35117 · doi:10.1007/BF02101705
[20] Pillet C.A., Wayne C.E.: Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations. J. Differ. Equ. 141(2), 310-326 (1997) · Zbl 0890.35016 · doi:10.1006/jdeq.1997.3345
[21] Reed, M.: Abstract Non-Linear Wave Equations. Lecture Notes in Mathematics, Vol. 507. Springer, Berlin, 1976 · Zbl 0317.35002
[22] Reed M., Simon B.: Methods of Modern Mathematical Physics, Vol. III. Academic Press, New York (1979) · Zbl 0405.47007
[23] Rodnianski I., Schlag W., Soffer A.: Dispersive analysis of charge transfer models. Commun. Pure Appl. Math. 58(2), 149-216 (2005) · Zbl 1130.81053 · doi:10.1002/cpa.20066
[24] Sigal I.M.: Nonlinear wave and Schrödinger equations. I: Instability of periodic and quasiperiodic solutions. Commun. Math. Phys. 153(2), 297-320 (1993) · Zbl 0780.35106 · doi:10.1007/BF02096645
[25] Soffer A., Weinstein M.I.: Multichannel nonlinear scattering for nonintegrable equations. Commun. Math. Phys. 133, 119-146 (1990) · Zbl 0721.35082 · doi:10.1007/BF02096557
[26] Soffer A., Weinstein M.I.: Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data. J. Differ. Equ. 98(2), 376-390 (1992) · Zbl 0795.35073 · doi:10.1016/0022-0396(92)90098-8
[27] Soffer A., Weinstein M.I.: Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 136, 9-74 (1999) · Zbl 0910.35107 · doi:10.1007/s002220050303
[28] Soffer A., Weinstein M.I.: Selection of the ground states for NLS equations. Rev. Math. Phys. 16(8), 977-1071 (2004) · Zbl 1111.81313 · doi:10.1142/S0129055X04002175
[29] Strauss, W.A.: Nonlinear invariant wave equations. Lecture Notes in Physics, Vol. 73. Springer, Berlin, 197-249, 1978
[30] 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 · doi:10.1002/cpa.3012
[31] Tsai T.-P.: Asymptotic dynamics of nonlinear Schrödinger equations with many bound states. J. Differ. Equ. 192(1), 225-282 (2003) · Zbl 1038.35128 · doi:10.1016/S0022-0396(03)00041-X
[32] Weinstein M.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16(3), 472-491 (1985) · Zbl 0583.35028 · doi:10.1137/0516034
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