## 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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### References:

 [1] Buslaev, V. S.; Perelman, G. S., Scattering for the nonlinear Schrödinger equations: states close to a soliton. St, Petersburg Math. J., 4, 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, 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, 419-475, (2003) · Zbl 1028.35139 [4] Cuccagna, S., Stabilization of solutions to nonlinear Schrödinger equations, Commun. Pure Appl. Math., 54, 1110-1145, (2001) · Zbl 1031.35129 [5] Cuccagna, S., On asymptotic stability of ground states of NLS, Rev. Math. Phys., 15, 877-903, (2003) · Zbl 1084.35089 [6] Cuccagna, S.; Mizumachi, T., On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations, Commun. Math. Phys., 284, 51-77, (2008) · Zbl 1155.35092 [7] Cuccagna, S., On asymptotic stability in 3D of kinks for the $${\phi^4}$$ model, Trans. AMS, 360, 2581-2614, (2008) · Zbl 1138.35062 [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 [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 [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, 321-367, (2006) · Zbl 1127.35054 [13] Kirr, E.; Zarnesku, A., On the asymptotic stability of bound states in 2D cubic Schrödinger equation, Commun. Math. Phys., 272, 443-468, (2007) · Zbl 1194.35416 [14] Komech, A.; Kopylova, E., Weighted energy decay for 1D Klein-Gordon equation, Commun. PDE, 35, 353-374, (2010) · Zbl 1190.35134 [15] Kopylov, S.A.: Private communication [16] Lions, J.L.: Quelques Mèthodes de Rèsolution des Problémes aux Limites non Linéaires. Paris, Dunod, 1969 [17] Merkli, M.; Sigal, I. M., A time-dependent theory of quantum resonances, Commun. Math. Phys., 201, 549-576, (1999) · Zbl 0934.47007 [18] Miller, J.; Weinstein, M., Asymptotic stability of solitary waves for the regularized long-wave equation, Commun. Pure Appl. Math., 49, 399-441, (1996) · Zbl 0854.35102 [19] Pego, R. L.; Weinstein, M. I., Asymptotic stability of solitary waves, Commun. Math. Phys., 164, 305-349, (1994) · Zbl 0805.35117 [20] Pillet, C. A.; Wayne, C. E., Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations, J. Differ. Equ., 141, 310-326, (1997) · Zbl 0890.35016 [21] Reed, M.: Abstract Non-Linear Wave Equations. Lecture Notes in Mathematics, Vol. 507. Springer, Berlin, 1976 [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, 149-216, (2005) · Zbl 1130.81053 [24] Sigal, I. M., Nonlinear wave and Schrödinger equations. I: Instability of periodic and quasiperiodic solutions, Commun. Math. Phys., 153, 297-320, (1993) · Zbl 0780.35106 [25] Soffer, A.; Weinstein, M. I., Multichannel nonlinear scattering for nonintegrable equations, Commun. Math. Phys., 133, 119-146, (1990) · Zbl 0721.35082 [26] Soffer, A.; Weinstein, M. I., Multichannel nonlinear scattering for nonintegrable equations. II, The case of anisotropic potentials and data. J. Differ. Equ., 98, 376-390, (1992) · Zbl 0795.35073 [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 [28] Soffer, A.; Weinstein, M. I., Selection of the ground states for NLS equations, Rev. Math. Phys., 16, 977-1071, (2004) · Zbl 1111.81313 [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, 153-216, (2002) · Zbl 1031.35137 [31] Tsai, T.-P., Asymptotic dynamics of nonlinear Schrödinger equations with many bound states, J. Differ. Equ., 192, 225-282, (2003) · Zbl 1038.35128 [32] Weinstein, M., Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16, 472-491, (1985) · Zbl 0583.35028
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