Invariant manifold of modified solitons for the perturbed sine-Gordon equation. (English) Zbl 1479.35746


35Q53 KdV equations (Korteweg-de Vries equations)
35L70 Second-order nonlinear hyperbolic equations
35C08 Soliton solutions
35B20 Perturbations in context of PDEs
35A24 Methods of ordinary differential equations applied to PDEs
35R01 PDEs on manifolds


Zbl 1442.35065
Full Text: DOI


[1] Alejo, M. A.; Corcho, A. J., Orbital stability of the black soliton for the quintic Gross-Pitaevskii equation (2020)
[2] Alejo, M. A.; Muñoz, C.; Palacios, J. M., On the asymptotic stability of the sine-Gordon kink in the energy space (2020)
[3] Benjamin, T. B., Applications of Leray-Schauder degree theory to problems of hydrodynamic stability, Math. Proc. Camb. Phil. Soc., 79, 373-392 (1976) · Zbl 0351.76054
[4] Bona, J., On the stability theory of solitary waves, Proc. R. Soc. A, 344, 363-374 (1975) · Zbl 0328.76016
[5] Buslaev, V. S.; Perel’man, G. S., On nonlinear scattering of states which are close to a soliton, Astérisque, 6, 49-63 (1992) · Zbl 0795.35111
[6] Deimling, K., Nonlinear Functional Analysis (1985), Berlin: Springer, Berlin · Zbl 0559.47040
[7] Farah, L. G.; Holmer, J.; Roudenko, S.; Yang, K., Asymptotic stability of solitary waves of the 3D quadratic Zakharov-Kuznetsov equation (2020)
[8] Frenkel, J.; Kontorova, T., On the theory of plastic deformation and twinning, Acad. Sci. USSR J. Phys., 1, 137-149 (1939) · JFM 65.1469.02
[9] Fröhlich, J.; Gustafson, S.; Jonsson, B. L G.; Sigal, I. M., Solitary wave dynamics in an external potential, Commun. Math. Phys., 250, 613-642 (2004) · Zbl 1075.35075
[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] Holmer, J.; Lin, Q., Phase-driven interaction of widely separated nonlinear Schrödinger solitons, J. Hyperbolic Differ. Equ., 9, 511-543 (2012) · Zbl 1256.35137
[12] Holmer, J., Dynamics of KdV solitons in the presence of a slowly varying potential, Int. Math. Res. Not., 23, 5367-5397 (2011) · Zbl 1247.35132
[13] Holmer, J.; Zworski, M.; Zworski, M., Slow soliton interaction with delta impurities, J. Mod. Dyn., 1, 689-718 (2007) · Zbl 1137.35060
[14] Holmer, J.; Zworski, M., Soliton interaction with slowly varying potentials, Int. Math. Res. Not., 2008 (2008) · Zbl 1147.35084
[15] Imaykin, V.; Komech, A.; Vainberg, B., Scattering of solitons for coupled wave-particle equations, J. Math. Anal. Appl., 389, 713-740 (2012) · Zbl 1235.35068
[16] Inoue, M.; Chung, S. G., Bion dissociation in sine-Gordon system, J. Phys. Soc. Japan, 46, 1594-1601 (1979)
[17] Jonsson, B. L G.; Fröhlich, J.; Gustafson, S.; Sigal, I. M., Long time motion of NLS solitary waves in a confining potential, Ann. Henri Poincaré, 7, 621-660 (2006) · Zbl 1100.81019
[18] Kivshar, Y. S.; Malomed, B. A., Dynamics of solitons in nearly integrable systems, Rev. Mod. Phys., 61, 763-915 (1989)
[19] Kowalczyk, M.; Martel, Y.; Muñoz, C., Kink dynamics in the ϕ^4 model: asymptotic stability for odd perturbations in the energy space, J. Am. Math. Soc., 30, 769-798 (2017) · Zbl 1387.35419
[20] Kopylova, E., Asymptotic stability of solitons for nonlinear hyperbolic equations, Habilitation Thesis (2015) · Zbl 1400.35201
[21] Mashkin, T., Stability of the solitary manifold of the sine-Gordon equation, Ph.D. Thesis (2016)
[22] Mashkin, T., Stability of the solitary manifold of the perturbed sine-Gordon equation, J. Math. Anal. Appl., 486 (2020) · Zbl 1442.35065
[23] Mashkin, T., Solitons in the presence of a small, slowly varying perturbation, Appl. Anal., 99, 2258-2279 (2020) · Zbl 1448.35450
[24] Mikeska, H. J., Solitons in a one-dimensional magnet with an easy plane, J. Phys. C: Solid State Phys., 11, L29 (1978)
[25] Natali, F.; de Loreno, G., Odd periodic waves for some Klein-Gordon type equations: existence and stability (2020)
[26] Skyrme, T. H R., Particle states of a quantized meson field, Proc. R. Soc. A, 262, 237-245 (1961) · Zbl 0099.43605
[27] Stuart, D. M A., Perturbation theory for kinds, Commun. Math. Phys., 149, 433-462 (1992) · Zbl 0756.35084
[28] Stuart, D. M A., Sine Gordon notes (2012)
[29] Soffer, A.; Weinstein, M. I., Multichannel nonlinear scattering for nonintegrable equations, Commun. Math. Phys., 133, 119-146 (1990) · Zbl 0721.35082
[30] Weinstein, M. I., Lyapunov stability of ground states of nonlinear dispersive evolution equations, Commun. Pure Appl. Math., 39, 51-67 (1986) · Zbl 0594.35005
[31] Zhang, L.; Huang, L.; Qiu, X. M., Josephson junction dynamics in the presence of microresistors and an AC drive, J. Phys.: Condens. Matter., 7, 353 (1995)
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.