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

Mashkin, Timur

Nonlinearity 34, No. 10, 6930-6962 (2021).

MSC:

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

Keywords:

soliton; iteration scheme; invariant manifold; sine-Gordon equation

Citations:

Zbl 1442.35065

References:

[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)